Quantum Chemistry in the Age of Quantum Computing
Open accessAbstract
Practical challenges in simulating quantum systems on classical computers have been widely recognized in the quantum physics and quantum chemistry communities over the past century. Although many approximation methods have been introduced, the complexity of quantum mechanics remains hard to appease. The advent of quantum computation brings new pathways to navigate this challenging and complex landscape. By manipulating quantum states of matter and taking advantage of their unique features such as superposition and entanglement, quantum computers promise to efficiently deliver accurate results for many important problems in quantum chemistry, such as the electronic structure of molecules. In the past two decades, significant advances have been made in developing algorithms and physical hardware for quantum computing, heralding a revolution in simulation of quantum systems. This Review provides an overview of the algorithms and results that are relevant for quantum chemistry. The intended audience is both quantum chemists who seek to learn more about quantum computing and quantum computing researchers who would like to explore applications in quantum chemistry.
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TL;DR
This review surveys how quantum computing can address quantum chemistry, covering classical limitations, computational complexity, fault-tolerant simulation algorithms, and NISQ-era variational methods. It argues quantum computers may efficiently handle wave functions and dynamics that are classically intractable, while near-term VQE remains heuristic and fault-tolerant QPE is longer-term.
Key findings
- Classical quantum chemistry is limited by exponential wave-function scaling; quantum computers can natively represent and evolve entangled states, and QPE with Hamiltonian simulation can estimate spectra, but requires fault tolerance. The review states that even 30–100 error-corrected qubits could compute H₂O and LiH ground-state energies beyond classical accuracy.
- For NISQ devices, VQE and hybrid quantum-classical algorithms are the leading approach. Hamiltonian averaging measurement cost scales roughly as \(M^2/\epsilon^2\); for electronic structure in local/Gaussian bases the expected scaling is \(O(N^6/\epsilon^2)\) versus the formal \(O(N^8/\epsilon^2)\), while a plane-wave dual basis reduces Hamiltonian terms to \(O(N^2)\) versus \(O(N^4)\) for Gaussians and gives a measurement bound of \(O(N^4/\epsilon^2)\).
- Fault-tolerant algorithm resource estimates improved substantially: Trotter steps can be implemented with \(O(N)\) depth using a fermionic swap network; LCU costs are \(\tilde{O}(N^5)\) in second quantization and \(\tilde{O}(\eta^2 N^3)\) in first quantization; interaction-picture simulations achieve \(O(N^{11/3})\) and \(O(\eta \log N)\); Babbush et al. report T complexity \(O(N+\log 1/\epsilon)\) for Hamiltonian simulation and \(O(N^3/\epsilon + N^2 \log 1/\epsilon)\) for energy estimation.
- Experimental VQE demonstrations include photonic HeH⁺ with 2 qubits, superconducting H₂ with 2 qubits, LiH with 4 qubits, BeH₂ with 6 qubits, and ion-trap H₂/LiH with 2/3 qubits; the review’s H₂ minimal-basis VQE simulation was numerically equal to FCI.
Methods
- Design: a review article, not primary research, synthesizing algorithms and results relevant to quantum chemistry on quantum computers for both quantum chemists and quantum computing researchers.
- Data or sample: no new experimental sample; it surveys cited literature and includes a pedagogical H₂ minimal-basis (STO-6G) VQE example, mapped via Bravyi-Kitaev to a four-qubit then two-qubit Hamiltonian.
- Main analysis: narrative and taxonomic review across classical quantum chemistry limitations, computational complexity, fault-tolerant algorithms (Hamiltonian simulation, QPE, QFT), NISQ hybrid algorithms (VQE, VQS), and non-gate-based methods (adiabatic, linear optics).
Limitations
- Stated: NISQ/VQE and related hybrid algorithms are heuristic with no performance guarantees; proper functioning must be established case-by-case and needs more hardware testing. Fault-tolerant algorithms require scalable error-corrected quantum computers, widely believed to be decades away. Complexity results are worst-case and may not reflect practical difficulty.
- Evident: as a review, it reports no original data or new benchmarks; resource estimates and experimental results are drawn from cited works, and many timeline claims are forecasts.
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Claims the paper makes
0 of 446 cited sources checked · 0 supported · 0 flagged
- Findingreasoning
Using recipes for molecular electronic structure implementations, implementing e^{-iHt} for an n-qubit Hamiltonian takes time polynomial in n on a quantum computer.
“Using such recipes, implementing e -iHt for an n -qubit Hamiltonian takes time polynomial in n on a quantum computer”
Quote in the paperSubstantial claim - Findingreasoning
Upon measurement of the second qubit register, the chance of obtaining the ground-state phase is dictated by |β_0|^2.
“Upon measurement of the second qubit register, one would like to maximize the chance of obtaining | ˜ ϕ 0 〉 , which is dictated by | β 0 | 2 .”
Quote in the paperSubstantial claim - Findingreasoning
Applying phase estimation on an initial state |φ〉|0〉 yields ∑_j β_j |ψ_j〉|φ̃_j〉.
“Applying phase estimation on the initial state | φ 〉| 0 〉 yields ∑ j β j | ψ j 〉| ˜ ϕ j 〉 .”
Quote in the paperSubstantial claim - Findingcitation
Initial simulations show that the imaginary-time variational quantum simulator method is competitive with the traditional gradient descent approach of the variational quantum eigensolver.
“Initial simulations show that this method is competitive with the traditional gradient descent approach of the variational quantum eigensolver.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [365] Olson, J.; Sim, S.; Cao, Y. Implementation of CUSP Using Cirq. https://github.com/zapatacomputing/cusp_cirq_demo, 2018 (accessed Nov 5, 2018).
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- Findingown data
In a simulation of the VQE algorithm for molecular hydrogen in the minimal basis (STO-6G), the computed ground state energies along the energy surface were numerically equal to the corresponding FCI energy values.
“This is observed for our small example of molecular hydrogen, shown in Figure 13a, in which a simulation of the VQE algorithm was able to compute ground state energies along the energy surface that were numerically equal to the corresponding FCI energy values.”
- Also stated in: § C.5 Variational quantum eigensolver for quantum chemistry · p. 150 · plain
“At each bond length, the energy computed using VQE is numerically equal to that of the FCI method.”
Quote in the paperNot a substantial claim - Also stated in: § C.5 Variational quantum eigensolver for quantum chemistry · p. 150 · plain
- Premisecitation
Hamiltonian simulation can be used with the quantum phase estimation algorithm to efficiently obtain the eigen-energies and eigenstates of a quantum system.
“the ability to efficiently perform Hamiltonian simulation is used in conjunction with another technique known as the quantum phase estimation algorithm (QPEA) 42 for efficiently obtaining the eigen-energies and eigenstates of a quantum system 38 .”
Quote in the paperNot a substantial claim0 of 2 sources checked - Interpretationreasoning
A quantum computer can potentially circumvent the exponential cost roadblock by being quantum mechanical itself.
“A quantum computer, as envisioned by Feynman and Manin, can potentially circumvent the roadblock of exponential cost by being quantum mechanical itself.”
Quote in the paperSubstantial claim - Premisereasoning
For even moderately large systems, explicitly maintaining the full wave function is intractable on a classical computer because the dimension of the wave function grows exponentially.
“In these cases, the exponential growth of the dimension of the wave function makes manipulation and storage very inefficient on a classical computer. Indeed, for even moderately large systems it is already intractable to explicitly maintain the full wave function.”
Quote in the paperSubstantial claim - Interpretationreasoning
Quantum computers can aid quantum chemistry computations that require an explicit representation of the wave function, either because of a high accuracy requirement of simulated properties or because of a high degree of entanglement in the system.
“For now, at least, we are confident that quantum computers can aid those quantum chemistry computations that require an explicit representation of the wave function, either because of a high accuracy requirement of simulated properties or because of a high degree of entanglement in the system.”
Quote in the paperSubstantial claim - Premisereasoning
Simulation of a general quantum mechanical system has proven to be difficult.
“However, simulation of a general quantum mechanical system has proven to be difficult.”
Quote in the paperSubstantial claim - Interpretationreasoning
Replacing specific classical subroutines with quantum computations can improve the accuracy and tractability of chemical predictions.
“the replacement of specific classical subroutines with quantum computations can improve the accuracy and tractability of chemical predictions.”
Quote in the paperNot a substantial claim - Interpretationreasoning
By off-loading a portion of the computation to a classical computer, a variational quantum algorithm is a far more practical alternative to its QPEA-based counterparts.
“By off-loading a portion of the computation to a classical computer, a variational quantum algorithm is a far more practical alternative to its QPEA-based counterparts.”
Quote in the paperNot a substantial claim - Premisereasoning
The FCI wave function cannot be efficiently manipulated and stored on a classical computer.
“Such wave function cannot be efficiently manipulated and stored on a classical computer.”
Quote in the paperSubstantial claim - Interpretationcitation
The breakdown of truncated coupled cluster can be generalized to any level of truncation of the excitation operator when strong correlation is exhibited by the Hamiltonian.
“This behavior can be generalized to any level of truncation of the excitation operator when strong correlation is exhibited by the hamiltonian 81 .”
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- Premisecitation
Coupled cluster with only single and double excitations breaks down for molecules in dissociation.
“The breakdown of coupled cluster with only single and double excitations for molecules in dissociation 80 is well-documented.”
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- Interpretationcitation
Entanglement-based tensor product methods are closer to the formulation of gate-model quantum computation than the classical methods discussed so far.
“Compared to the classical methods discussed so far, entanglementbased tensor product methods 100 , which have been applied to study strongly correlated many-body physics and quantum chemistry, are closer to the formulation of gate-model quantum computation.”
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- Premisecitation
Quantum simulation in first quantization can be readily applied to non-Born-Oppenheimer instances.
“Quantum simulation in first quantization can be readily applied to non-BOA instances as well 75,76 .”
Quote in the paperNot a substantial claim0 of 2 sources checked- cites [75] Aharonov, D.; van Dam, W.; Kempe, J.; Landau, Z.; Lloyd, S.; Regev, O. Adiabatic Quantum Computation Is Equivalent to Standard Quantum Computation. 45th Annual IEEE Symposium on Foundations of Computer Science , 2004.
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- cites [76] Aaronson, S.; Arkhipov, A. The Computational Complexity of Linear Optics. Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing , New York, 2011; pp 333–342. source
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- Premisecitation
An exact solution of the time-independent non-relativistic Schrödinger equation for molecular systems quickly becomes intractable due to exponential growth of the wave function size with particle number.
“However, an exact solution quickly becomes intractable due to the exponential growth of the size of the wave function with the particle number 69 .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [69] source
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- Premisecitation
Several algorithms have been proposed for Monte Carlo integration on a quantum computer that outperform their classical counterparts.
“Measurement outcomes from the output of a quantum circuit are probabilistic and several algorithms have been proposed for Monte Carlo integration on a quantum computer that outperform their classical counterparts 91 .”
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- Premisecitation
Exact diagonalization requires storing coefficients of all relevant determinants, which becomes prohibitive even for medium-sized molecules.
“This comes at the cost of storing the coefficients of all (relevant) determinants, which becomes prohibitive even for medium-sized molecules 94,95 .”
Quote in the paperSubstantial claim0 of 2 sources checked - Interpretationreasoning
On a quantum computer, if the cluster operator can be represented as a low-depth quantum circuit, it is efficient to prepare a unitary coupled cluster wave function.
“If the cluster operator can be represented as a low-depth quantum circuit, it is efficient to prepare a unitary coupled cluster wave function.”
Quote in the paperSubstantial claim - Premisenone given
Coupled cluster's non-Hermitian formulation makes the obtained energy non-variational.
“While coupled cluster theory is a promising method for quantum chemistry, its nonHermitian formulation makes the obtained energy non-variational.”
Quote in the paperSubstantial claim - Interpretationcitation
The quantum phase estimation algorithm is the natural translation of the FCI procedure to quantum computers.
“The Quantum Phase Estimation (QPE) algorithm 38,99 is the natural translation of the FCI procedure to quantum computers 45 .”
Quote in the paperNot a substantial claim0 of 3 sources checked- cites [38] Grover, L. K. A Fast Quantum Mechanical Algorithm for Database Search. Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing ; New York, 1996; pp 212–219. source
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- cites [99] source
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- cites [45] source
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- Premisecitation
DMRG applied to quantum chemistry can generally handle large active spaces (approximately 50 molecular orbitals).
“DMRG applied to quantum chemistry can generally handle large active spaces ( ∼ 50 MOs 115 ).”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [115] source
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- Interpretationnone given
Linear response theory is one of the most cost-effective ways to obtain time-dependent properties without explicitly doing time propagation.
“One of the most cost effective ways to obtain time-dependent properties without explicitly doing time propagation is the first-order perturbation approach or linear response theory.”
Quote in the paperNot a substantial claim - Premisecitation
Some biochemical processes such as conformational transition in hemoglobin occur on a time scale many times the current limit of MD calculations.
“Some processes in biochemistry such as conformational transition in hemoglobin occur on a time scale that is many times the current limit of MD calculations 149 .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [149] source
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- Interpretationreasoning
After longer time evolution, errors due to inherent approximations in tensor-based methods accumulate and lead to loss of accuracy and a rapidly growing bond dimension.
“After longer time evolution, errors due to the inherent approximations accumulate and lead to loss of accuracy and a rapidly growing bond dimension.”
Quote in the paperNot a substantial claim - Interpretationreasoning
These problems pose a practical limit on the applicability of tensor network states for quantum chemistry.
“However, these problems pose a practical limit on the applicability of tensor network states (TNS) for quantum chemistry.”
Quote in the paperSubstantial claim - Premisenone given
DMRG wave functions at a low bond dimension miss a sizable amount of dynamic correlation important in quantum chemistry.
“However, DMRG wave functions at a low bond dimension miss a sizable amount of the dynamic correlation that is important in quantum chemistry.”
Quote in the paperNot a substantial claim - Premisecitation
DMRG handles much larger active spaces than conventional active-space methods such as complete active space self-consistent field.
“It handles much larger active spaces than conventional active-space methods such as complete active space self consistent field 116 .”
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- Interpretationcitation
Low bond dimension MPS are not candidates for quantum speedup because slightly entangled quantum computations may be efficiently simulated on a classical computer using the MPS formalism.
“It must be noted that low bond dimension MPS are not candidates for quantum speedup as slightly entangled quantum computations may be efficiently simulated on a classical computer using the MPS formalism 138 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [138] source
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- Premisecitation
Tensor product methods have proved to be efficient and accurate as multireference computational methods.
“On the contrary, tensor product methods 100 have proved to be efficient and accurate as multireference computational methods.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [100] source
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- Premisecitation
For linear or quasi-linear molecular systems, active spaces of a hundred molecular orbitals are reachable with DMRG.
“In case of linear (or quasi-linear) molecular systems, active spaces of a hundred of MOs are reachable 118 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [118] source
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- Interpretationnone given
The most severe limitation of MD is the limited amount of time propagation it currently supports.
“The most severe problem is the limited amount of time propagation that MD currently supports.”
Quote in the paperNot a substantial claim - Premisecitation
Series methods (Taylor, Padé, Chebyshev) are accurate for small matrices but prohibitively expensive for larger systems.
“The Taylor expansion, Pad´ e approximant, and Chebyshev approximant are series methods based on powers of the Hamiltonian 150,151 to approximate the evolution operator. They are accurate for small matrices but prohibitively expensive for larger systems.”
Quote in the paperNot a substantial claim0 of 2 sources checked - Interpretationnone given
All of the quantum algorithms mentioned in this subsection rely on the possibility of fault-tolerant quantum computers in order to outperform their classical counterparts.
“All of the quantum algorithms mentioned in this subsection rely on the possibility of fault-tolerant quantum computers in order to outperform any of their classical counterparts.”
Quote in the paperSubstantial claim - Interpretationreasoning
To obtain a specific eigenstate of a Hamiltonian such as the ground state via phase estimation, an appropriate initial state reasonably close to the target eigenstate is needed.
“It should be noted that in the case where the objective is a specific eigenstate of the Hamiltonian such as the ground state | ψ 0 〉 , an appropriate initial state | φ 0 〉 , reasonably close to | ψ 0 〉 is needed.”
Quote in the paperSubstantial claim - Premisecitation
Classical simulation of time evolution is tractable when the dynamical sign problem can be effectively suppressed.
“the dynamical sign problem can be effectively suppressed 185 .”
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- Premisecitation
The sequences of operations yielded from phase estimation are often too deep to be feasible on today's NISQ devices.
“However, an important technical point is that the sequences of operations (or quantum circuits, see Section 4.1.3) yielded from phase estimation are often too deep to be feasible on today's NISQ devices 51 .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [51] source
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- Premisecitation
For molecular electronic structure problems, recipes for implementing each e^{-iH_j t} have been explicitly specified.
“For molecular electronic structure problems the recipes for such implementations have been explicitly specified 186 .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [186] source
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- Premisecitation
There are already proposals for NISQ algorithms based on variational principles.
“Indeed, there are already proposals 200 for NISQ algorithms based on variational principles 201,202 .”
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- cites [201] source
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- cites [202] Berry, D. W.; Childs, A. M.; Cleve, R.; Kothari, R.; Somma, R. D. Exponential Improvement in Precision for Simulating Sparse Hamiltonians. Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing , New York, 2014; pp 283–292. source
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- Interpretationreasoning
For NISQ devices, the hybrid quantum-classical algorithm paradigm is a more practical framework for developing heuristics to solve eigenstate and energy-level problems.
“For NISQ devices, the recently proposed paradigm of hybrid quantum-classical algorithm is a more practical framework for developing heuristics to solve the same problems of eigenstates and energy levels.”
Quote in the paperSubstantial claim - Premisenone given
A classical computer takes time at least ∼2^n to implement e^{-iHt} for an n-qubit Hamiltonian.
“while a classical computer takes time at least ∼ 2 n .”
Quote in the paperNot a substantial claim - Premisecitation
On a fault-tolerant quantum computer, there are efficient methods for implementing an approximation of e^{-iHt} for arbitrary t and a broad class of Hamiltonians H.
“On a (fault-tolerant) quantum computer, there are efficient methods for implementing an approximation of e -iHt for arbitrary t and a broad class of Hamiltonians H .”
Quote in the paperSubstantial claim - Premisecitation
A similar interaction-picture technique has been applied in quantum chemistry to produce the first quantum simulation algorithm with cost scaling sublinearly in the number of basis functions.
“A similar technique has also been applied in quantum chemistry 198 to produce the first quantum simulation algorithm of cost scaling sublinearly in the number of basis functions.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [198] Kivlichan, I. D.; Gidney, C.; Berry, D. W.; Wiebe, N.; McClean, J.; Sun, W.; Jiang, Z.; Rubin, N.; Fowler, A.; Aspuru-Guzik, A.; Babbush, R.; Neven, H. Improved Fault Tolerant Quantum Simulation of Condensed-Phase Correlated Electrons via Trotterization. 2019, arXiv:1902.10673 [quant-ph]; https://arxiv.org/abs/1902.10673. source
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- Interpretationreasoning
To obtain the ground-state energy in a time duration scaling polynomially in the number of qubits n, |β_0|^2 must be at least as large as an inverse polynomial in n.
“In order to obtain the ground state energy in a time duration that scales polynomially in the number of qubits n acted on by the Hamiltonian H , we require that | β 0 | 2 be at least as large as an inverse polynomial in n .”
Quote in the paperSubstantial claim - Premisenone given
Propagating an initial wave function in time by calculating |ψ(t)〉 = e^{-iHt}|ψ_0〉 is in general hard on a classical computer.
“Propagating an initial wave function | ψ 0 〉 in time by calculating | ψ ( t ) 〉 = e -iHt | ψ 0 〉 is in general hard on a classical computer.”
Quote in the paperSubstantial claim - Interpretationreasoning
The availability of methods for preparing initial wave functions and well-studied molecular electronic Hamiltonians gives hope that quantum computers can use quantum phase estimation to efficiently compute the spectrum of quantum systems to accuracy comparable to FCI.
“Both ingredients give hope that quantum computers can take advantage of quantum phase estimation to efficiently compute the spectrum of quantum systems to accuracy comparable to FCI.”
Quote in the paperSubstantial claim - Premisecitation
Finding the ground state energy of many simple locally interacting systems is QMA-complete.
“Most notably, it has been shown that finding the ground state energy of many simple locally interacting systems is QMA -complete 207,208 .”
Quote in the paperNot a substantial claim0 of 2 sources checked - Premisecitation
Some recently developed paradigms for Hamiltonian simulation achieve exponential improvements in precision compared with their predecessors.
“Some of the recently developed paradigms for Hamiltonian simulation 187-190 achieve exponential improvements in precision compared with their predecessors 187,188 .”
Quote in the paperNot a substantial claim0 of 4 sources checked - Premisecitation
One can transform the Hamiltonian expressed in terms of fermionic operators into a linear combination of tensor products of Pauli matrices that preserves the fermionic anticommutation relations.
“While there are several methods for accomplishing this transform 180,181 , all result in a Hamiltonian that is a linear combination of tensor products of matrices { I, X, Y, Z } with I = ( 1 0 0 1 ) in such a way that preserves the fermionic anticommutation relations.”
Quote in the paperSubstantial claim0 of 2 sources checked - Premisecitation
The phase estimation algorithm transforms an eigenstate into a state that encodes an approximation of the phase φ_j/(2π), with approximation error from using a finite qubit register.
“For an eigenstate | ψ j 〉 , the phase estimation 1 algorithm transforms the state on a quantum computer into”
- Also stated in: § 2.3.1 Statics: phase estimation and variational quantum eigensolvers · p. 30 · plain
“where | ˜ ϕ j 〉 is a state that encodes an approximation of the phase ϕ j / (2 π ). The approximation error comes from the attempt to capture the value of ϕ j , which is a continuous variable, with a finite qubit register and hence a finite-dimensional quantum system.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [1] Nielsen M. A., 2010
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- Also stated in: § 2.3.1 Statics: phase estimation and variational quantum eigensolvers · p. 30 · plain
- Premisecitation
Techniques for simulating time-dependent Hamiltonians using Dyson series have been proposed, with scaling matching the time-independent case.
“In parallel with previous methods for simulating time-independent Hamiltonians 188 yielding near-optimal cost scaling, techniques for simulating time-dependent Hamiltonians using Dyson series 199 have also been recently proposed, with scaling matching the time-independent case 188 .”
Quote in the paperNot a substantial claim0 of 2 sources checked - Premisecitation
Recently developed Hamiltonian simulation paradigms have been applied to quantum chemistry, yielding a similar exponential improvement.
“They have also been applied to quantum chemistry 191,192 yielding a similar exponential improvement.”
Quote in the paperNot a substantial claim0 of 2 sources checked - Premisenone given
Classical simulation of time evolution is tractable when the system is small enough for the Hamiltonian to be treatable with explicit methods.
“the system is small enough for the Hamiltonian to be treatable with explicit methods;”
Quote in the paperNot a substantial claim - Premisecitation
Classical simulation of time evolution is tractable when efficient approximation of the wave function is possible using tensor networks or neural networks.
“efficient approximation of the wave function is possible using, for instance, tensor networks, as in t-DMRG, or more recently neural networks 184 ;”
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- Premisecitation
The ability to manipulate a discretization of the wave function directly opened a floodgate of new results in quantum algorithms for quantum chemistry.
“the ability to manipulate a discretization of the wave function directly opened a floodgate of new results in quantum algorithms for quantum chemistry 37,40,41,75,233 .”
Quote in the paperNot a substantial claim0 of 3 sources checked - Premisereasoning
Solving the eigenvalue problem exactly is much more challenging than Hamiltonian simulation.
“Solving such an eigenvalue problem exactly is a much more challenging problem than the Hamiltonian simulation.”
Quote in the paperNot a substantial claim - Premisecitation
Berry et al. showed that one query to select(V) can be implemented with d^2 calls to O_loc and O_val.
“Berry et al. 187 showed that one query to select( V ) can be implemented with d 2 calls to O loc and O val .”
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- Premisecitation
The NP-completeness of Hartree-Fock persists even if one restricts to translationally invariant systems.
“The NP -completeness has also been proven to persist even if one restricts to translationally invariant systems 213 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [213] Babbush, R.; Berry, D. W.; McClean, J. R.; Neven, H. Quantum Simulation of Chemistry With Sublinear Scaling to the Continuum. 2018, arXiv:1807.09802 [quant-ph]; https://arxiv.org/abs/1807.09802.
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- Premisecitation
The N-representability problem has been proven QMA-complete.
“the problem of checking whether a given set of two-electron reduced density matrices (2RDM) is consistent with a single many-electron wave function, known as the N -representability problem , has also been proven QMA -complete 66 .”
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- Interpretationcitation
Assuming the number of terms l in the Hamiltonian scales polynomially with system size N, the scaling of the number of gates is polynomial in N, τ, and 1/ε.
“Assuming that the number of terms ℓ in the Hamiltonian scales polynomially with the system size N (such as the number of particles or the number of qubits), then the scaling of the number of gates is polynomial in N , τ , and 1 /ϵ .”
Quote in the paperSubstantial claim- The authors state this as an assumption: The number of terms ℓ in the Hamiltonian scales polynomially with the system size N.
- Premisecitation
With respect to a desired accuracy and duration of time evolution, the number of quantum gates grows polynomially in system size, time, and inverse precision.
“with respect to a desired accuracy and duration of the time evolution, the number of quantum gates (roughly, computation time) grows polynomially in system size, time, and inverse-precision.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [37] source
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- Interpretationreasoning
Quantum computers can provide an exponential improvement in memory resources for quantum simulation compared to a naive approach using a wave function.
“In this setting, quantum computers can provide an exponential improvement in memory resources for quantum simulation compared to a naive approach using a wave function.”
Quote in the paperSubstantial claim - Premisereasoning
These defects limit the length of programs that can be accurately executed.
“These defects limit the length of the programs that can be accurately executed.”
Quote in the paperSubstantial claim - Interpretationreasoning
Any efficient Hamiltonian simulation algorithm requires the simulated Hamiltonian to have a special structure.
“Accordingly, Lloyd's result, or any other efficient Hamiltonian simulation algorithm, requires the simulated Hamiltonian to have a special structure.”
Quote in the paperNot a substantial claim - Premisecitation
For a Hamiltonian of l terms describing k-body interactions, U(t) can be approximated within error ε by a sequence of O(lτ^2/ε) elementary quantum gates, where τ=||H||t.
“The formal result from Lloyd's paper is that, for a Hamiltonian of ℓ terms describing k -body interactions, U ( t ) can be approximated within error ϵ by a sequence of O ( ℓτ 2 /ϵ ) elementary quantum gates 250 , where τ = || H || t .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [250] source
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- Premisenone given
The general Hamiltonian eigenvalue problem is QMA-complete.
“In fact, this problem known to be QMA -complete, for a general Hamiltonian, as discussed in Section 3.”
Quote in the paperNot a substantial claim - Premisecitation
For an arbitrary Hamiltonian H, the number of elementary gates needed to construct U(t)=e^{-iHt} grows exponentially with the number of qubits.
“For an arbitrary Hamiltonian H , the number of elementary gates needed to construct U ( t ) = e -iHt grows exponentially with the number of qubits 1 .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [1] Nielsen M. A., 2010
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- Interpretationreasoning
Obtaining a classical description of a quantum state stored in quantum bits would require repeated quantum simulation and tomography, which essentially eliminates all the savings.
“Obtaining a classical description of the quantum state would require repeated quantum simulation and tomography which essentially eliminates all the savings.”
Quote in the paperSubstantial claim- Rests on an assumption: The amplitudes of a wave function stored in quantum bits cannot be efficiently accessed.
- Premisecitation
Solving the Hartree-Fock method has been shown to be NP-complete.
“Solving the Hartree-Fock (HF) method, which intends to find a single Slater determinant that best approximates the true wave function, has been shown to be NP -complete 65 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [65] Colless, J.; Ramasesh, V.; Dahlen, D.; Blok, M.; McClean, J.; Carter, J.; de Jong, W. A.; Siddiqi, I. Implementing a Variational Quantum Eigensolver Using Superconducting Qubits. Quantum Information and Measurement 2017 , 2017; p QF6A.2. source
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- Premisecitation
Many physically relevant Hamiltonians are sparse or even local.
“As such many physically-relevant Hamiltonians are sparse 254 or even local 186,255 .”
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The wave function can be efficiently evolved in time for many physically realistic Hamiltonians.
“It is important that the wave function can be efficiently evolved in time for many 'physically realistic' Hamiltonians 37 .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [37] source
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- Premisecitation
Given enough time, a fault-tolerant quantum computer is able to execute an arbitrarily long circuit without sacrificing accuracy.
“Given enough time, a faulttolerant quantum computer is able to execute an arbitrarily long circuit without sacrificing accuracy.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [239] source
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- Premisecitation
Lloyd's work gave a rigorous upper bound on the computational time for quantum simulation.
“the work of Lloyd gave a rigorous upper bound on the computational time for quantum simulation.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [37] source
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- Interpretationreasoning
The Trotter decomposition for local-interaction Hamiltonian simulation is efficient.
“Thus, the Trotter decomposition for local-interaction Hamiltonian simulation is efficient.”
Quote in the paperSubstantial claim - Interpretationreasoning
It is possible to obtain multiple eigenstates and eigenenergies of H by running QPEA repeatedly with an appropriately chosen initial state |φ>.
“Therefore, it is possible to obtain multiple eigenstates and eigenenergies of H by running QPEA repeatedly with an appropriately chosen initial state | φ 〉 , highlighting the importance of the state preparation step, as discussed in Section 4.2.1.”
Quote in the paperSubstantial claim - Premisecitation
The number of ancilla qubits can be greatly reduced, while maintaining the same precision, by using the iterative quantum phase estimation algorithm (IPEA).
“As detailed by Aspuru-Guzik et al. 45 and Dobˇ s´ ıˇ cek et al. 275 , the number of ancilla qubits can be greatly reduced, while maintaining the same precision by using the iterative quantum phase estimation algorithm (IPEA).”
Quote in the paperNot a substantial claim0 of 2 sources checked - Premisecitation
Product-formula based algorithms required fewer resources than quantum signal processing in a numerical simulation for systems between 10 and 100 qubits.
“In particular, product-formula based algorithms required fewer resources than quantum signal processing in a numerical simulation for systems between 10 and 100 qubits.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [272] source
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- Premisecitation
The Bayesian estimation approach to quantum phase estimation has recently been shown experimentally, demonstrating its robustness to noise and decoherence.
“This Bayesian estimation approach to quantum phase estimation has recently been shown experimentally 278 , demonstrating its robustness to noise and decoherence.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [278] source
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- Premisecitation
Low and Chuang developed algorithms based on quantum signal processing and qubitization that achieve provably optimal asymptotic scaling in query complexity O(dτ_max + log(1/ε) log log(1/ε)), achieving additive approximation error.
“In 2016, Low and Chuang developed algorithms based on quantum signal processing 190 and qubitization 189 which achieve this provably optimal asymptotic scaling in query complexity of O ( dτ max + log(1 /ϵ ) log log(1 /ϵ ) ), achieving the sought-after 'additive' approximation error.”
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Berry et al. improved the antisymmetrization procedure by using a sorting network and optimizing components of the algorithm.
“Recently, Berry et al. 279 improved the antisymmetrization procedure by using a sorting network and optimizing components of the algorithm.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [279] source
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- Premisecitation
Childs showed that given oracle access to a Hamiltonian with eigenvalues λ, one can construct a unitary quantum walk operator with eigenvalues e±i arcsin λ.
“Childs 257 showed that given oracle access to a Hamiltonian with eigenvalues λ , one can construct a unitary quantum walk operator with eigenvalues e ± i arcsin λ .”
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- Premisereasoning
The probability of obtaining the eigenstate λ_m is |a_m|^2, the squared overlap between the eigenstate and the prepared superposition.
“The probability of obtaining the eigenstate λ m is | a m | 2 , i.e. the squared overlap between the eigenstate with the prepared superposition.”
Quote in the paperSubstantial claim - Premisecitation
If the norm of A is much larger than the norm of B, say for a diagonally dominant matrix, interaction picture simulations can lead to an exponential improvement over existing techniques.
“If the norm of A is much larger than the norm of B , say for a diagonally dominant matrix, interaction picture simulations can lead to an exponential improvement over existing techniques.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [197] source
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- Interpretationreasoning
Roughly speaking, each ancilla adds another digit of accuracy.
“Roughly speaking, each ancilla adds another digit of accuracy.”
Quote in the paperNot a substantial claim - Premisecitation
Berry et al. presented an algorithm in 2014 that exponentially improved the dependence on precision.
“A breakthrough in quantum simulation methods came in 2014 when Berry et al. presented an algorithm that exponentially improved the dependence on precision 187 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [187] source
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- Premisereasoning
These Hamiltonians can be implemented without the use of oracles requiring O(TL(n+log L) log(T/ε) log log(T/ε)) gates, where n is the number of qubits, L is the number of terms, and T = sum α_j.
“These Hamiltonians can be implemented without the use of oracles requiring O ( T L ( n +log L ) log ( T/ϵ ) log log ( T/ϵ ) ) gates, where n is the number of the qubits in the simulated system, L is the number number of terms in the Hamiltonian and T = ∑ L -1 j =0 α j .”
Quote in the paperNot a substantial claim - Premisecitation
By more sophisticated Trotter-Suzuki decompositions, the approximation error can be made to depend on progressively higher orders in τ.
“First, by more sophisticated Trotter-Suzuki decompositions 261 , the approximation error can be made to depend on progressively-higher orders in τ 196,260,261 .”
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Antisymmetrization can be enforced by reversing a suitable sorting algorithm and uncomputing ancillas to make the procedure reversible.
“Antisymmetrization can be enforced by reversing a suitable sorting algorithm and uncomputing ancillas to make the procedure reversible 286 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [286] source
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- Premisecitation
By considering more favorable schemes for Hamiltonian decomposition, most modern-day simulation algorithms scale in terms of τ_max = t||H||_max instead of τ.
“Second, by considering more favorable schemes for Hamiltonian decomposition 256,257 , most modern-day simulation algorithms scale in terms of τ max = t ‖ H ‖ max instead of τ , where ‖ H ‖ max is the absolute value of the largest entry of H as a matrix in the computational basis.”
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QPEA gives an exponential advantage to many of the algorithms reviewed in the next section.
“It gives an exponential advantage to many of the algorithms reviewed in the next section.”
Quote in the paperSubstantial claim0 of 2 sources checked- cites [231] Wocjan, P.; Zhang, S. Several Natural BQP-Complete Problems. 2006, arXiv:quant-ph/0606179; https://arxiv.org/abs/quant-ph/0606179.
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- cites [263] source
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- Premisecitation
Given a unitary operator e^{iΦ} and an approximate eigenstate prepared on the quantum computer, QPEA is used to efficiently output a binary representation of the corresponding phase Φ.
“Given a unitary operator e i Φ and an approximate eigenstate of that operator prepared on the quantum computer, the quantum phase estimation algorithm is used to efficiently output a binary representation of the corresponding phase Φ.”
Quote in the paperSubstantial claim0 of 2 sources checked- cites [231] Wocjan, P.; Zhang, S. Several Natural BQP-Complete Problems. 2006, arXiv:quant-ph/0606179; https://arxiv.org/abs/quant-ph/0606179.
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- cites [263] source
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- Premisecitation
A recent study showed that different algorithms fare better in different regimes.
“A recent study 272 showed that different algorithms fare better in different regimes.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [272] source
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- Premisenone given
During simulation, the kinetic and interaction/potential energy terms can both be implemented diagonally by applying the quantum Fourier transform to change between the momentum and position basis.
“During simulation, the kinetic and interaction/potential energy terms can both be implemented diagonally by applying the quantum Fourier transform to change between the momentum and position basis.”
Quote in the paperNot a substantial claim - Premisereasoning
After execution of QPEA, measurement of the first register will yield a binary approximation of the eigenenergy λ_m with high probability.
“After execution of the algorithm, measurement of the first register will yield a binary approximation of the eigenenergy λ m with a high probability.”
Quote in the paperSubstantial claim - Premisecitation
With probability at least 1-ε, the estimation of λ_m is accurate to (T - ceil(log(2 + 1/(2ε)))) bits.
“It can be shown that with probability at least 1 -ϵ , the estimation of λ m is accurate to ( T -⌈ log ( 2 + 1 2 ϵ ) ⌉ ) bits 1 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [1] Nielsen M. A., 2010
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- Interpretationreasoning
Advances in quantum algorithms for Hamiltonian simulation lead to improvements in the performance of QPEA.
“Accordingly, advances in quantum algorithms for Hamiltonian simulation lead to improvements in the performance of QPEA.”
Quote in the paperNot a substantial claim - Premisecitation
This technique leads to quantum speedups both for first- and second-quantized problem formulations.
“It has been shown that this technique leads to quantum speedups both for first- and second-quantized problem formulations 40,75,254,284 .”
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- cites [75] Aharonov, D.; van Dam, W.; Kempe, J.; Landau, Z.; Lloyd, S.; Regev, O. Adiabatic Quantum Computation Is Equivalent to Standard Quantum Computation. 45th Annual IEEE Symposium on Foundations of Computer Science , 2004.
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- cites [254] source
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- Interpretationnone given
Despite recent improvements in the resource estimates for fully-quantum chemistry algorithms such as the quantum phase estimation, the number of gates and circuit depth required for their implementation surpasses the capabilities of existing and near-term quantum devices.
“Despite recent improvements in the resource estimates for fully-quantum chemistry algorithms such as the quantum phase estimation, the number of gates and circuit depth required for their implementation surpasses the capabilities of existing and near-term quantum devices.”
Quote in the paperNot a substantial claim - Premisecitation
Early LCU-based algorithms have been shown to scale better asymptotically as a function of molecule size than prior Trotter algorithms for quantum simulation.
“Importantly, these early LCU-based algorithms have been shown to scale better asymptotically as a function of molecule size than prior Trotter algorithms for quantum simulation.”
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It was demonstrated that the Trotter errors depend on the maximum nuclear charge rather than the number of spin-orbitals.
“Later, it was also demonstrated that the Trotter errors depend on the maximum nuclear charge rather than the number of spin-orbitals 50 .”
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- Premisecitation
Poulin et al. empirically studied the cost of simulation for real-world molecules, finding that the cost was closer to O(~N^6).
“Poulin et al. 49 empirically studied the cost of simulation for real-world molecules, finding that the cost was closer to O ( ∼ N 6 ).”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [49] source
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- Premisecitation
Early devices, also called 'pre-threshold' or NISQ devices, cannot support quantum error correction schemes, which require additional overhead.
“Furthermore, these early devices, also called 'pre-threshold' or NISQ 51 devices, cannot support quantum error correction schemes, which require additional overhead.”
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- Premisecitation
In second quantization, the number of gates required for LCU is Õ(N^5).
“In second quantization, the number of gates required is ˜ O ( N 5 ).”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [192] source
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- Premisecitation
In the adiabatic regime, the evolved state has support at least 1 - O(1/T) on the ground state.
“This is a so-called adiabatic regime where the evolved state has support at least 1 -O ( 1 T ) on the ground state 292 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [292] source
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- Premisecitation
Kinetic energy terms can be diagonalized using the quantum Fourier transform.
“Kinetic energy terms can be diagonalized using the quantum Fourier transform.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [273] Childs, A. M.; Kothari, R. Simulating Sparse Hamiltonians With Star Decompositions, Conference on Quantum Computation, Communication, and Cryptography , 2010; pp 94–103. source
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- Premisecitation
ASCI allows for the generation of a small number of determinants while accounting for at least 90% of the wave function.
“ASCI allows for the generation of a small number of determinants while accounting for at least 90% of the wave function.”
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It was shown that Trotter steps exactly N layers of gates deep (with N the number of spin-orbitals) could be performed even with the restriction of the qubits being on a line.
“Following this, it was shown that Trotter steps exactly N layers of gates deep (with N the number of spin-orbitals) could be performed even with the restriction of the qubits being on a line 302 .”
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- Premisecitation
Kivlichan et al. showed that single Trotter steps of the electronic structure Hamiltonian can be simulated in O(N) depth with only nearest-neighbor connectivity on a line.
“Kivlichan et al. further refined this to show that single Trotter steps of the electronic structure Hamiltonian can be simulated in O ( N ) depth with only nearest-neighbor connectivity on a line 302 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [302] source
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- Premisecitation
Crosson et al. demonstrated that this effect is common even for instances that do not exhibit noticeable symmetries.
“Crosson et al. 298 demonstrated that this effect is common even for instances that do not exhibit noticeable symmetries.”
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- Premisecitation
In first quantization, the number of gates required for LCU is Õ(η^2 N^3), where η is the number of particles.
“In first quantization, it is ˜ O ( η 2 N 3 ), where η is the number of particles in the system.”
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- Premisecitation
Incorporating state-of-the-art error correction methods, Babbush et al. show that the estimates for the number of gates required to estimate the ground state energy of FeMoCo is millions of times smaller than the number needed in the methods used earlier.
“Incorporating state-of-the-art error correction methods, they show that the estimates for the number of gates (which is dominated by the number of T -gates) required to estimate the ground state energy of FeMoCo is millions of times smaller than the number needed in the methods used earlier 305 .”
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- cites [305] Abrams, D. S. Quantum Algorithms. Ph.D. thesis, Massachusetts Institute of Technology, 1999.
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- Premisecitation
If the Hamiltonian and its first derivative are upper-bounded by a constant, a good approximation of the ground state can be reached for evolution time T = O(1/γ^2) where γ is the minimum gap (except for pathological cases).
“If the Hamiltonian and its first derivative are upper-bounded by a constant, a good approximation of the ground state can be reached for evolution time T = O ( 1 γ 2 ) where γ is the minimum gap (except for pathological cases 291 ).”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [291] source
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- Interpretationcitation
Using arbitrary or unstructured ansatze will lead to poor convergence of the optimization routine.
“This implies that using arbitrary or unstructured ansatze will lead to poor convergence of the optimization routine.”
Quote in the paperSubstantial claim - Premisecitation
Even with modest quantum computers of 30-100 error-corrected qubits, ground state energy calculations of H2O and LiH could be carried out to an accuracy beyond that of classical computers.
“This work shows that even with modest quantum computers of 30-100 errorcorrected qubits, ground state energy calculations of H 2 O and LiH could be carried out to an accuracy beyond that of classical computers.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [45] source
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- Premisecitation
McClean et al. show that, for sufficiently random circuits, the variational parameter landscape is plagued by the overabundance of 'barren plateaus', large regions in which the cost function gradient is nearly zero.
“The authors show that, for sufficiently random circuits, the variational parameter landscape is plagued by the overabundance of 'barren plateaus', large regions in which the cost function gradient is nearly zero.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [321] source
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- Premisecitation
Aspuru-Guzik et al. first realized that quantum computers can be used for determining ground-state energies.
“Aspuru-Guzik et al. 45 first realized that quantum computers can be used for determining ground-state energies.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [45] source
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- Premisecitation
Symmetries in the spectrum can be exploited even when the gap is exponentially small.
“Symmetries in the spectrum can be exploited even when the gap is exponentially small 297 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [297] source
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- Premisereasoning
Toloui and Love's technique is more space efficient compared to second quantized techniques using Fock states.
“This technique is also more space efficient compared to second quantized techniques using Fock states.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [254] source
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- Interpretationcitation
It is not always necessary to reach the adiabatic regime to achieve a significant overlap with the ground state.
“Note that it is not always necessary to reach the adiabatic regime to achieve a significant overlap with the ground state.”
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Hardware heuristic ansatz approaches can be implemented more easily than existing problem-inspired ansatze on NISQ devices.
“Although these approaches can be implemented more easily than existing PMA on NISQ devices, to date, there lacks theoretical evidence one way or the other regarding their ability to efficiently approximate many-electron wave functions.”
Quote in the paperNot a substantial claim - Interpretationnone given
The scaling in the number of parameters of the UCCSD ansatz poses a challenge for typical optimization strategies, as the number of amplitudes can easily increase to the order of thousands for systems with tens of spin-orbitals and half filling.
“The scaling in the number of parameters of the UCCSD ansatz poses a challenge for typical optimization strategies, as the number of amplitudes can easily increase to the order of thousands for systems with tens of spin-orbitals and half filling.”
Quote in the paperSubstantial claim - Interpretationnone given
The Bogoliubov basis allows the application of UCC to Hamiltonians with pairing fields, extending the range of applicability of VQE to problems in condensed matter and nuclear physics.
“The latter allows for the application of UCC to Hamiltonians with pairing fields, extending the range of applicability of VQE to problems in condensed matter and nuclear physics.”
Quote in the paperNot a substantial claim - Premisecitation
UCC is not a practical ansatz in classical quantum chemistry because the BCH expansion of the similarity-transformed Hamiltonian does not terminate.
“However, UCC is not a practical ansatz in classical quantum chemistry because the BCH expansion of the similarity transformed Hamiltonian, e -( T -T † ) He T -T † , does not terminate 324 .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [324] source
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- Premisecitation
The Hamiltonian variational strategy can achieve good overlaps with the ground state for values of S varying between 3 and 20.
“The results show that this strategy can achieve good overlaps with the ground state for values of S varying between 3 and 20 (see Wecker et al. 325 ).”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [325] source
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- Premisenone given
Unitary operations such as e^{T-T†} are natural operations on quantum computers.
“Fortunately, unitary operations such as e T -T † are natural operations on quantum computers.”
Quote in the paperSubstantial claim - Premisenone given
In Gaussian-type orbitals, the number of non-negligible terms scales nearly quadratically for systems of sufficient size.
“In the case of Gaussian-type orbitals (GTOs), which are the most popular basis sets in quantum chemistry, the number of non-negligible terms scales nearly quadratically for systems of sufficient size, as explained in more detail in Section 5.1.2.”
Quote in the paperSubstantial claim - Premisereasoning
The O(N^7) scaling of traditional coupled-cluster methods limits their applicability to small molecules.
“Unfortunately the scaling of this approach, O ( N 7 ), limits its applicability to small molecules.”
Quote in the paperSubstantial claim - Premisereasoning
A serial implementation of UCCSD terms results in a scaling of the number of gates of O(fN^4), where f is the number of gates required for each term.
“Correspondingly, a serial implementation of these terms would result in a scaling of the number of gates of O ( fN 4 ), where f indicates the number of gates required for implementing each term.”
Quote in the paperNot a substantial claim - Premisecitation
When combined with the simplified representations of the Hamiltonian, the Hamiltonian variational method can be used to construct variational circuits with depth equal to the number of spin-orbitals for a linear qubit architecture, using the fermionic swap network.
“When combined with the simplified representations of the Hamiltonian described in the previous section 301 , the Hamiltonian variational method can be used to construct variational circuits with depth equal to the number of spin-orbitals for a linear qubit architecture, using the fermionic swap network 302 .”
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The accuracy of the exact implementation of the UCCSD exponential operator is virtually the same as that obtained with Trotterized UCCSD for different Trotter numbers.
“The effect of the number of Trotter steps has been also investigated, indicating that the accuracy of the exact implementation of the UCCSD exponential operator is virtually the same as the one obtained with Trotterized UCCSD for different Trotter numbers.”
Quote in the paperNot a substantial claim - Premisereasoning
The O(N^4) scaling of the Hamiltonian variational ansatz depth might vary depending on the analytical properties of the basis sets employed to represent the Hamiltonian.
“This scaling might vary depending on the analytical properties of the basis sets employed to represent the Hamiltonian.”
Quote in the paperSubstantial claim - Premisecitation
The sizes at which the nearly quadratic scaling is observed depend on the nature of the system and the basis set and might correspond to tens to hundreds of angstroms.
“The sizes at which this scaling is observed, however, depends on the nature of the system and the basis set and might correspond to tens to hundreds of angstroms 79,306 .”
Quote in the paperSubstantial claim0 of 2 sources checked- cites [79] source
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- cites [306] Tubman, N. M.; Mejuto-Zaera, C.; Epstein, J. M.; Hait, D.; Levine, D. S.; Huggins, W.; Jiang, Z.; McClean, J. R.; Babbush, R.; Head-Gordon, M.; Whaley, K. B. Postponing the Orthogonality Catastrophe: Efficient State Preparation for Electronic Structure Simulations on Quantum Devices. 2018, arXiv:1809.05523 [quant-ph]; https://arxiv.org/abs/1809.05523.
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- Premisenone given
For quantum chemistry Hamiltonians, the depth of the Hamiltonian variational ansatz is directly related to the number of terms in the Hamiltonian, which scales formally as O(N^4).
“However, in the case of quantum chemistry Hamiltonians, the depth of the corresponding ansatz is directly related to the number of terms in the Hamiltonian, which scales formally as O ( N 4 ).”
Quote in the paperSubstantial claim - Interpretationreasoning
If the initial and target Hamiltonians are both expressed as a sum over the same Hamiltonian terms, the Hamiltonian variational evolution can be seen as analogous to a discretized adiabatic state preparation, with angles obtained by variational optimization instead of a predefined annealing schedule.
“If the initial and target Hamiltonian are both expressed as a sum over the same Hamiltonian terms, H i = ∑ j h i j O j , then the evolution in Equation 37 can be seen as analogous to a discretized adiabatic state preparation, where the angles of the rotations are obtained by a variational optimization instead of”
Quote in the paperNot a substantial claim - Premisecitation
For architectures with arbitrary connectivity that allow parallelization, strategies developed for Trotterized Hamiltonian evolution could save up to a linear factor in depth using a constant number of ancilla qubits when a Jordan-Wigner mapping is used.
“For architectures with arbitrary connectivity that allow parallelization, some strategies developed for Trotterized Hamiltonian evolution 49 could save up to a linear factor in depth using a constant number of ancilla qubits when a Jordan-Wigner mapping is used.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [49] source
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- Premisecitation
UCCSD has been demonstrated experimentally for the wave function of a deuterium nucleus using cloud quantum computing.
“for H 2 and LiH on an ion-trapped quantum computer 59 and for the wave function of a deuterium nucleus using cloud quantum computing 330 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [330] source
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- Premisecitation
The application of the UCC ansatz truncated up to single and double excitations has been investigated both theoretically and experimentally.
“Given the inherent advantage in implementing the ansatz as well as the high accuracies of CC methods, the application of the UCC ansatz truncated up to single and double excitations (UCCSD) has been investigated both theoretically and experimentally 52,53,59,325-330 .”
Quote in the paperNot a substantial claim0 of 9 sources checked- cites [52] Wiesner, S. Simulations of Many-Body Quantum Systems by a Quantum Computer. 1996, arXiv:quant-ph/9603028; https://arxiv.org/abs/quant-ph/9603028.
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- cites [53] source
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- cites [59] source
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- cites [325] source
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- cites [326] source
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- cites [327] source
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- cites [328] source
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- cites [329] source
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- cites [330] source
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- Premisecitation
Babbush et al. proposed using a dual form of the plane wave basis that diagonalizes the potential operator, leading to a representation of the electronic structure Hamiltonian with only O(N^2) terms.
“In particular, Babbush et al. have proposed to use a dual form of the plane wave basis 301 which diagonalizes the potential operator, leading to the following representation of the electronic structure Hamiltonian that has only O ( N 2 ) terms:”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [301] source
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- Premisecitation
The Hamiltonian variational approach is inspired by both adiabatic state preparation and the quantum approximate optimization algorithm.
“The Hamiltonian variational approach is inspired by both adiabatic state preparation (ASP) and the quantum approximate optimization algorithm (QAOA) 331,332 .”
Quote in the paperNot a substantial claim0 of 2 sources checked - Premisecitation
UCCSD has been demonstrated experimentally for H2 and LiH on an ion-trapped quantum computer.
“for H 2 and LiH on an ion-trapped quantum computer 59 and for the wave function of a deuterium nucleus using cloud quantum computing 330 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [59] source
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- Premisecitation
Different Trotter approximations to the UCC operator constitute a family of CC-inspired ansatze that can be implemented on a quantum computer using standard circuit compilation techniques for exponentiating Pauli matrices.
“The different Trotter approximations to the UCC operator constitute a family of CC-inspired ansatze that can be implemented on a quantum computer using standard circuit compilation techniques for exponentiating Pauli matrices 1 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [1] Nielsen M. A., 2010
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- Premisereasoning
Considering parallelization, the final LDCA circuit depth is (10L+8)ceil(N/2)+4 when accounting for U†Bog and the initial round of single-qubit X gates.
“Considering parallelization, the final circuit depth is (10 L +8) ⌈ N 2 ⌉ + 4 when we account for U † Bog and the initial round of single-qubit X gates.”
Quote in the paperNot a substantial claim - Premisecitation
The final circuit for preparing fermionic Gaussian states from the covariance matrix can be parallelized such that its depth scales only as O(N).
“The final circuit can be parallelized such that its depth scales only as O ( N ).”
Quote in the paperNot a substantial claim - Premisecitation
More recently, the LDCA has been applied to lithium hydride, showing superior accuracy compared to hardware efficient ansatze.
“More recently, the LDCA has been applied to lithium hydride 339 , showing superior accuracy compared to hardware efficient ansatze.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [339] Kandala, A.; Temme, K.; Corcoles, A. D.; Mezzacapo, A.; Chow, J. M.; Gambetta, J. M. Extending the Computational Reach of a Noisy Superconducting Quantum Processor. 2018, arXiv:1805.04492 [quant-ph]; https://arxiv.org/abs/1805.04492.
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- Premisecitation
The operator U can be decomposed into a sequence of Givens rotations by applying a QR decomposition scheme to u, generating a circuit composed of two-qubit gates corresponding to these Givens rotations applied on neighboring qubits.
“U can be decomposed into a sequence of Givens rotations by applying a QR decomposition scheme to u , generating a circuit composed of two-qubit gates corresponding to these Givens rotations applied on neighboring qubits 302,335 .”
Quote in the paperNot a substantial claim0 of 2 sources checked- cites [302] source
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- cites [335] Novo, L.; Berry, D. W. Improved Hamiltonian Simulation via a Truncated Taylor Series and Corrections. 2016, arXiv:1611.10033 [quant-ph]; https://arxiv.org/abs/1611.10033.
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- Premisecitation
The results demonstrated the ability of LDCA to describe the ground states of these systems in both weakly-correlated and strongly-correlated regimes.
“The results demonstrated the ability of LDCA to describe the ground states of these systems in both weakly-correlated and strongly-correlated regimes.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [326] source
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- Premisecitation
These hardware efficient ansatz implementations have incorporated the use of active spaces and methodologies for reducing the number of qubits in the Hamiltonian.
“These implementations have incorporated the use of active spaces and methodologies for reducing the number of qubits in the Hamiltonian 343 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [343] source
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- Premisecitation
The circuit implementing the transformation W can be parallelized in depth O(N).
“This circuit can be parallelized in depth O ( N ) 335 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [335] Novo, L.; Berry, D. W. Improved Hamiltonian Simulation via a Truncated Taylor Series and Corrections. 2016, arXiv:1611.10033 [quant-ph]; https://arxiv.org/abs/1611.10033.
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- Premisenone given
When the Jordan-Wigner transformation acts on the terms (a†p aq + a†q ap), it produces terms of locality |p-q|+1, which has significant overhead when mapped to a circuit.
“The main insight of this proposal is that when the Jordan-Wigner transformation acts on the terms ( a † p a q + a † q a p ), it produces terms of locality | p -q | + 1, which has significant overhead when mapped to a circuit.”
Quote in the paperSubstantial claim - Premisecitation
Methods for preparing Slater determinants and general fermionic Gaussian states employ a quadratic number of gates.
“Methods for preparing Slater determinants and general FGSs employ a quadratic number of gates 302,326,335,336 .”
Quote in the paperNot a substantial claim0 of 4 sources checked- cites [302] source
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- cites [326] source
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- cites [335] Novo, L.; Berry, D. W. Improved Hamiltonian Simulation via a Truncated Taylor Series and Corrections. 2016, arXiv:1611.10033 [quant-ph]; https://arxiv.org/abs/1611.10033.
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- cites [336] source
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- Premisecitation
Jiang et al. proposed a method that specifies fermionic Gaussian states as the corresponding quadratic fermionic Hamiltonian for which the target states are ground states.
“Jiang et al. 335 proposed a method that specifies the FGSs as the corresponding quadratic fermionic Hamiltonian for which the target FGSs are ground states.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [335] Novo, L.; Berry, D. W. Improved Hamiltonian Simulation via a Truncated Taylor Series and Corrections. 2016, arXiv:1611.10033 [quant-ph]; https://arxiv.org/abs/1611.10033.
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- Premisenone given
Rotations by the interaction and repulsion terms between orbitals with indexes p and q could be implemented together with the f-SWAP gates, forming the fermionic simulation gate.
“Furthermore, rotations by the interaction and repulsion terms between orbitals with indexes p and q could be implemented together with the f-SWAP gates, forming the so-called fermionic simulation gate ,”
Quote in the paperNot a substantial claim - Premisecitation
The accuracy of hardware efficient ansatze has been tested both experimentally and numerically for small molecular systems including H2, LiH, BeH2 and H2O.
“The accuracy of HEA have been tested both experimentally 56 and numerically 327 for small molecular systems including H 2 , LiH, BeH 2 and H 2 O.”
Quote in the paperNot a substantial claim0 of 2 sources checked - Interpretationnone given
The linear scaling of the depth of LDCA makes it a viable option for implementation on near-term NISQ devices.
“The linear scaling of the depth of LDCA makes it a viable option for implementation on near-term NISQ devices.”
Quote in the paperNot a substantial claim - Premisenone given
Using matrix manipulation on W, the corresponding circuit implementing the transformation W can be decomposed into a sequence of (N-1)N/2 Givens rotations applied on neighboring qubits and N particle-hole transformations implemented as single qubit rotations.
“Using matrix manipulation on W , the corresponding circuit implementing this transformation, W , can be decomposed into a sequence of ( N -1) N/ 2 Givens rotations applied on neighboring qubits and N particle-hole transformations implemented as single qubit rotations.”
Quote in the paperSubstantial claim - Premisenone given
Numerical studies of LDCA have shown great promise.
“Numerical studies of LDCA have shown great promise.”
Quote in the paperNot a substantial claim - Premisecitation
Dallaire-Demers et al. proposed a method that prepares fermionic Gaussian states specified by their corresponding covariance matrix.
“Independently, Dallaire-Demers et al. 326 proposed a method that prepares FGSs specified by their corresponding covariance matrix.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [326] source
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- Premisenone given
When mapped to qubit operators, the Hamiltonian in Equation 44 is equivalent to a local Hamiltonian in the Jordan-Wigner representation.
“When mapped to qubit operators, the Hamiltonian in Equation 44 is equivalent to the following local Hamiltonian in the Jordan-Wigner representation:”
Quote in the paperNot a substantial claim - Premisenone given
In the particular case of the Fermi-Hubbard Hamiltonian, the scaling is further reduced to O(√N).
“In the particular case of the Fermi-Hubbard Hamiltonian, the scaling is further reduced to O ( √ N ).”
Quote in the paperNot a substantial claim - Premisecitation
In general, to approach the accuracy of methods like UCCSD, the number of hardware efficient ansatz layers has to grow rapidly, even to an extent that the number of parameters approaches or surpasses the size of the Hilbert space.
“In general, in order to approach the accuracy of methods like UCCSD, the number of layers has to grow rapidly, even to an extent that the number of parameters approaches or surpasses the size of the Hilbert space 327 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [327] source
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- Premisereasoning
An L-cycle LDCA circuit has 5L(N-1)ceil(N/2)+N variational angles, with the extra term arising from the round of Z rotations.
“Since each cycle has ⌈ N 2 ⌉ layers, an L -cycle circuit has 5 L ( N -1) ⌈ N 2 ⌉ + N variational angles, the extra term arising from the round of Z rotations.”
Quote in the paperNot a substantial claim - Premisenone given
The set of these ground states lives on only a one-dimensional manifold in the four-qubit space.
“However, the set of these ground states lives on only a one-dimensional manifold in the fourqubit space.”
Quote in the paperNot a substantial claim - Interpretationnone given
Unlike PEA, where each k-local Pauli operator requires O(k) circuit depth, Hamiltonian averaging only adds a constant factor to the state preparation procedure, assuming parallel rotation and orbital readout is available.
“Unlike the PEA approach, where each k -local Pauli operator requires O ( k ) circuit depth for the implementation, Hamiltonian averaging only adds a constant factor to the state preparation procedure, assuming parallel”
Quote in the paperNot a substantial claim- The authors state this as an assumption: Parallel rotation and orbital readout is available for Hamiltonian averaging.
- Premisereasoning
In general, the number of state preparations and measurement cycles required to converge the energy to a precision ε scales roughly as M^2/ε^2, where M is the number of terms in the Pauli decomposition of the Hamiltonian.
“In general, the number of state preparations and measurement cycles required to converge the energy to a precision ϵ will scale roughly as M 2 /ϵ 2 , where M is the number of terms in the Pauli decomposition of the Hamiltonian.”
Quote in the paperNot a substantial claim- The authors state this as an assumption: All Hamiltonian terms are measured independently, i.e. Cov(〈O_i〉, 〈O_j〉) = 0 for all i ≠ j.
- Premisecitation
CUSP uses an existing VQE circuit ansatz together with a quantum autoencoder to generate a new ansatz that is shorter in gate depth.
“CUSP uses an existing VQE circuit ansatz, together with a quantum autoencoder, to generate a new ansatz that is shorter in gate depth and hence less vulernable to noise on a NISQ computer, resulting in more accurate VQE energies.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [345] source
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- Premisecitation
Finding a unitary mapping that coherently maps the four-qubit state to a single qubit enables a generative modeling approach called Compressed Unsupervised State Preparation that explores the latent space to find a shorter pathway to a desired ground state.
“The payoff for finding such a unitary mapping is that it enables a type of generative modeling approach called Compressed Unsupervised State Preparation (CUSP) that explores the latent space to find a shorter pathway to a desired ground state 345 .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [345] source
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- Premisecitation
The quantum autoencoder is an HQC algorithm that encodes a set of quantum states using a fewer number of quantum bits.
“Specifically, the quantum autoencoder 344 is an HQC algorithm that encodes a set of quantum states using a fewer number of quantum bits, as shown in Figure 8.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [344] source
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- Premisecitation
In practice, it has been observed that δ = 10^-10 is more than sufficient to retain quantitatively accurate results in molecular calculation with GTOs.
“In practice, it has been observed that δ = 10 -10 is more than sufficient to retain quantitatively accurate results in molecular calculation with GTOs 79,306,347 .”
Quote in the paperNot a substantial claim0 of 3 sources checked- cites [79] source
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- cites [306] Tubman, N. M.; Mejuto-Zaera, C.; Epstein, J. M.; Hait, D.; Levine, D. S.; Huggins, W.; Jiang, Z.; McClean, J. R.; Babbush, R.; Head-Gordon, M.; Whaley, K. B. Postponing the Orthogonality Catastrophe: Efficient State Preparation for Electronic Structure Simulations on Quantum Devices. 2018, arXiv:1809.05523 [quant-ph]; https://arxiv.org/abs/1809.05523.
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- cites [347] source
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- Premisecitation
Compressed Unsupervised State Preparation (CUSP) explores the latent space to find a shorter pathway to a desired ground state.
“Compressed Unsupervised State Preparation (CUSP) that explores the latent space to find a shorter pathway to a desired ground state 345 .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [345] source
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- Interpretationreasoning
There can be an advantage in preparing more complicated initial states that are closer to the final state.
“While improving circuit ansatze help reduce the overhead in the depth of a quantum circuit after the quantum computer has been initialized, there can be an advantage in preparing more complicated initial states that are closer to the final state.”
Quote in the paperNot a substantial claim - Interpretationreasoning
This analysis justifies the strategy of neglecting Hamiltonian terms that are under a certain threshold δ, such as the maximum δ ≤ ε/N_r, with N_r being the number of removed terms.
“Moreover, this analysis justifies the intuitive strategy of neglecting Hamiltonian terms that are under a certain threshold δ , such as the maximum δ ≤ ϵ N r , with N r being the number of removed terms.”
Quote in the paperSubstantial claim - Interpretationreasoning
An initial state that is already highly entangled could be advantageous to preparing a desired ground state.
“Thus, an initial state that is already highly entangled could be advantageous to preparing a desired ground state.”
Quote in the paperNot a substantial claim - Interpretationreasoning
The quantum information corresponding to this four-qubit state can be represented by a single qubit.
“This means that the quantum information corresponding to this four-qubit state can be represented by a single qubit.”
Quote in the paperSubstantial claim - Interpretationreasoning
Since in finite molecules and bulk materials η ∈ O(N), the number of measurements does not scale worse than O(N^4/ε^2) for this representation of the Hamiltonian.
“Since in finite molecules and bulk materials, η ∈ O ( N ), this shows that the number of measurements does not scale worse than O ( N 4 /ϵ 2 ) for this representation of the Hamiltonian.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [301] source
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- The authors state this as an assumption: The bound on the number of measurements for the plane wave dual basis assumes a neutral system and a fixed density.
- Rests on an assumption: In finite molecules and bulk materials, the number of electrons η is in O(N).
- Interpretationreasoning
The requirement of full coherence and quantum error correction for PEA is unattainable on NISQ devices because the operations available on these devices are too error-prone to achieve useful quantum error correction.
“This requirement is unattainable on NISQ devices because the operations available on these devices are too error-prone to achieve useful quantum error correction.”
Quote in the paperSubstantial claim - Premisereasoning
Imaginary-time evolution is useful for preparing a thermal state at temperature T with respect to the Hamiltonian.
“Imaginary-time evolution is useful for preparing a thermal state at temperature T with respect to the Hamiltonian: ρ ( T ) = e -H/T / tr( e -H/T ).”
Quote in the paperSubstantial claim - Interpretationreasoning
With approximate preparation of a Hamiltonian's ground state, the imaginary-time variational quantum simulator can be used as an alternative to the variational quantum eigensolver algorithm.
“With approximate preparation of a Hamiltonian's ground state, this algorithm can be used as an alternative to the variational quantum eigensolver algorithm.”
Quote in the paperSubstantial claim - Premisereasoning
If the system is initialized in the completely mixed state I/d, then the imaginary-time evolution of τ = 1/(2T) will yield a thermal state.
“If the system is initialized in the completely mixed state I /d , then the imaginary-time evolution of τ = 1 / (2 T ) will yield a thermal state e -τH · I d · e -τH ∝ e -H/T .”
Quote in the paperSubstantial claim - Premisecitation
In the variational quantum simulator, the entries of the matrix M and vector V can be estimated from measurements made on the quantum computer.
“The crux of the algorithm is that the entries of the matrix M and vector ⃗ V can be estimated from measurements made on the quantum computer.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [200] source
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- Premisecitation
The variational quantum simulator technique has been developed further for problems beyond quantum simulation.
“The variational quantum simulator technique has been developed further for problems beyond quantum simulation.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [365] Olson, J.; Sim, S.; Cao, Y. Implementation of CUSP Using Cirq. https://github.com/zapatacomputing/cusp_cirq_demo, 2018 (accessed Nov 5, 2018).
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- Premisecitation
The VQS version of determining excited state energies iteratively builds up the spectrum starting with the ground state energy.
“Similar to the excited-state energy methods described in Section 5.1.5, the VQS version of determining excited state energies iteratively builds up the spectrum starting with the ground state energy.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [339] Kandala, A.; Temme, K.; Corcoles, A. D.; Mezzacapo, A.; Chow, J. M.; Gambetta, J. M. Extending the Computational Reach of a Noisy Superconducting Quantum Processor. 2018, arXiv:1805.04492 [quant-ph]; https://arxiv.org/abs/1805.04492.
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- Premisenone given
Certain many-body terms in the Hamiltonian, such as three- and four-body terms, are difficult to implement experimentally.
“For the second issue, certain many-body terms in the Hamiltonian, such as three- and fourbody terms X 1 Z 2 X 3 and Y 1 Z 2 Y 3 Z 4 , are difficult to implement experimentally.”
Quote in the paperSubstantial claim - Premisereasoning
Preparation of a thermal state can be used for preparing approximations to the ground state of the Hamiltonian by taking the limit T → 0.
“Preparation of a thermal state can be used for preparing approximations to the ground state of the Hamiltonian by taking the limit T → 0, since, for a non-degenerate Hamiltonian | ψ gs 〉〈 ψ gs | = lim T → 0 e -H/T / tr( e -H/T ).”
Quote in the paperSubstantial claim - Premisecitation
In McArdle et al. 365, a variant of the variational quantum simulator is used for preparing certain useful quantum states.
“In McArdle et al. 365 a variant of the variational quantum simulator is used for preparing certain useful quantum states.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [365] Olson, J.; Sim, S.; Cao, Y. Implementation of CUSP Using Cirq. https://github.com/zapatacomputing/cusp_cirq_demo, 2018 (accessed Nov 5, 2018).
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- Premisenone given
The Hamiltonian must be changed slowly enough relative to the transition energy of the first excited state so that non-adiabatic transitions do not drive population out of the ground state.
“Additionally, the Hamiltonian must be changed slowly enough relative to the transition energy of the first excited state (the energy gap) so that non-adiabatic transitions do not drive population out of the ground state.”
Quote in the paperNot a substantial claim - Premisecitation
In quantum chemistry, research has focused on the analog simulation of molecules by mapping the electronic structure problem to a Hamiltonian of spins with additional coupling terms.
“In the context of quantum chemistry, research has focused on the analog simulation of molecules by mapping the electronic structure problem to a Hamiltonian of spins with additional coupling terms 366 , although proposals exist for simulating chemical reactions 367-369 and protein folding simulations 370,371 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [366] source
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- Premisecitation
A variant of the imaginary-time variational simulator has been proposed to determine excited state energies of Hamiltonians.
“Just as variants of VQE have been developed for determining excited state energies of Hamiltonians 320,360 , a variant of the imaginary-time variational simulator has been proposed to do the same 339 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [339] Kandala, A.; Temme, K.; Corcoles, A. D.; Mezzacapo, A.; Chow, J. M.; Gambetta, J. M. Extending the Computational Reach of a Noisy Superconducting Quantum Processor. 2018, arXiv:1805.04492 [quant-ph]; https://arxiv.org/abs/1805.04492.
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- Premisecitation
The Hamiltonian of molecular hydrogen in the minimal basis mapped to a four-qubit system using the Bravyi-Kitaev transformation takes a specific form.
“The Hamiltonian of molecular hydrogen in the minimal basis mapped to a four-qubit system, using the Bravyi-Kitaev transformation, takes the following form 181,366 :”
Quote in the paperNot a substantial claim0 of 2 sources checked - Premisecitation
In the VQS version for determining excited states, the penalty terms added to the Hamiltonian alter the imaginary-time dynamics, rather than the cost-function that is optimized.
“The difference in the VQS version for determining excited states is that these penalty terms added to the Hamiltonian alter the imaginary-time dynamics, rather than the cost-function that is optimized.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [339] Kandala, A.; Temme, K.; Corcoles, A. D.; Mezzacapo, A.; Chow, J. M.; Gambetta, J. M. Extending the Computational Reach of a Noisy Superconducting Quantum Processor. 2018, arXiv:1805.04492 [quant-ph]; https://arxiv.org/abs/1805.04492.
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- Premisecitation
The time evolution in the variational quantum simulator algorithm can be replaced by imaginary-time evolution e^{-i(-iτ)H}=e^{-τH}.
“The key insight of this paper is that the time evolution in the variational quantum simulator algorithm can be replaced by imaginary-time evolution e -i ( -iτ ) H = e -τH .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [365] Olson, J.; Sim, S.; Cao, Y. Implementation of CUSP Using Cirq. https://github.com/zapatacomputing/cusp_cirq_demo, 2018 (accessed Nov 5, 2018).
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- Premisecitation
In both VQE excited-state methods and the VQS version, each step introduces a penalty term to the Hamiltonian that is proportional to the projector onto the previously-found eigenstate.
“Furthermore, in both approaches, each step introduces a penalty term to the Hamiltonian that is proportional to the projector onto the previously-found eigenstate.”
Quote in the paperNot a substantial claim0 of 3 sources checked- cites [320] source
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- cites [360] source
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- cites [339] Kandala, A.; Temme, K.; Corcoles, A. D.; Mezzacapo, A.; Chow, J. M.; Gambetta, J. M. Extending the Computational Reach of a Noisy Superconducting Quantum Processor. 2018, arXiv:1805.04492 [quant-ph]; https://arxiv.org/abs/1805.04492.
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- Interpretationnone given
The information of RDMs measured in the VQE procedure can be used to connect the VQE algorithm with complementary classical algorithms for the calculation of other properties or energy refinement.
“The information of the RDMs measured in the VQE procedure can be used to connect the VQE algorithm with complementary classical algorithms for the calculation of other properties or energy refinement.”
Quote in the paperNot a substantial claim - Premisecitation
n-valence electron perturbation theory requires measurement of the 3-RDM and 4-RDM in order to compute the corresponding energy correction.
“For example, n -valence electron perturbation theory 361 requires measurement of the 3-RDM and and 4-RDM in order to compute the corresponding energy correction.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [361] source
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- Premisecitation
The first method for estimating electronic ground state energies, proposed by Aspuru-Guzik et al., uses the quantum phase estimation algorithm to make quantum measurements of the energy of a trial wave function.
“The first method, proposed by Aspuru-Guzik et al. 45 , uses the quantum phase estimation algorithm to make quantum measurements of the energy of a trial wave function, as described in Section 4.1.2.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [45] source
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- Premisecitation
Clements et al. 375 is the first instance of an experimental FCP of a small molecule, tropolone.
“The paper by Clements et al. 375 is the first instance of an experimental FCP of a small molecule, tropolone.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [375] source
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- Premisenone given
Although gadgetization schemes have greatly improved over the last decade, they provide limited tunability over the couplings that are required to construct the problem Hamiltonian.
“Although gadgetization schemes have greatly improved over the last decade, they provide limited tunability over the couplings that are required to construct the problem Hamiltonian.”
Quote in the paperSubstantial claim - Premisecitation
Techniques exist to reduce many-body terms to two-body terms by adding ancilla qubits and encoding the correct couplings in the lowest eigenstate of a different Hamiltonian, called gadgetization.
“Techniques exist to reduce them to two-body terms by adding ancilla qubits and encoding the correct couplings in the lowest eigenstate of a different Hamiltonian, which is commonly called gadgetization 42,208,372-374 .”
Quote in the paperNot a substantial claim0 of 5 sources checked- cites [42] source
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- cites [208] source
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- cites [372] Guerreschi, G. G.; Smelyanskiy, M. Practical Optimization for Hybrid Quantum Classical Algorithms. 2017, arXiv:1701.01450 [quant-ph]; https://arxiv.org/abs/1701.01450.
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- cites [373] Kennedy, J. Encyclopedia of Machine Learning ; Springer, 2011; pp 760–766. source
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- cites [374] source
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- Premisecitation
The variational quantum eigensolver has become the focus of state-of-the-art quantum experiments.
“This algorithm, known as the variational quantum eigensolver, as described in Section 5.1 has become the focus of the state-of-the-art quantum experiments 55,56 .”
Quote in the paperNot a substantial claim0 of 2 sources checked - Premisecitation
The improved algorithm accounts for the unavoidable effects of experimental imperfections.
“The improved algorithm accounts for the unavoidable effects of experimental imperfections.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [375] source
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- Premisecitation
Variational ansatze borrowed from classical quantum chemistry such as the unitary coupled cluster have informed state preparation techniques in several quantum algorithms.
“Variational ansatze borrowed from classical quantum chemistry such as the unitary coupled cluster have informed state preparation techniques 329 in several quantum algorithms.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [329] source
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- Premisecitation
The first realization of the linear-optics protocol for simulating the Franck-Condon profile was introduced by Huh and Yung.
“The first realization of this protocol was introduced by Huh and Yung 228 and targeted the simulation of the Franck-Condon profile.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [228] source
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- Premisenone given
This work and its successors are the only examples of native simulation of bosonic chemical systems on a quantum device to date.
“This work and its successors 375,376 are the only examples of native simulation of bosonic chemical systems on a quantum device to date.”
Quote in the paperNot a substantial claim0 of 2 sources checked - Interpretationreasoning
Quantum computers provide a solution to the problem of inaccuracies in post-Hartree-Fock calculations by naturally handling wave functions that span the full Hilbert space to estimate energies, electric polarization, magnetic dipoles, reduced density matrices, etc.
“Quantum computers provide a solution to this problem by naturally handling wave functions that span the full Hilbert space to estimate energies, electric polarization, magnetic dipoles, reduced density matrices, etc.”
Quote in the paperNot a substantial claim - Premisecitation
The modification of the optical apparatus stems from the unitary transformation that encodes the molecular problem.
“This modification of the optical apparatus stems from the unitary transformation (Equation 19) that encodes the molecular problem 220 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [220] source
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- Premisecitation
The N-representability problems and the 2-RDM constraints inspired the methods in Rubin et al. 346 for improving expectation value estimates in VQE.
“Finally, the Nrepresentability problems and the 2-RDM constraints inspired the methods in Rubin et al. 346 for improving expectation value estimates in VQE.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [346] source
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- Premisecitation
The ability to test performance on real quantum devices facilitates the development of heuristic quantum algorithms.
“In particular, the ability to test performance on real quantum devices facilitates the development of heuristic quantum algorithms 52,200,320,326,331,344,382 .”
Quote in the paperNot a substantial claim0 of 7 sources checked- cites [52] Wiesner, S. Simulations of Many-Body Quantum Systems by a Quantum Computer. 1996, arXiv:quant-ph/9603028; https://arxiv.org/abs/quant-ph/9603028.
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- cites [200] source
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- cites [320] source
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- cites [326] source
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- cites [331] Veis, L.; Pittner, J. Advances in Chemical Physics ; John Wiley & Sons Inc., 2014; pp 107–136. source
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- cites [344] source
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- cites [382] source
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- Interpretationreasoning
Quantum error correction would enable the use of various Hamiltonian simulation algorithms for quantum chemistry.
“Regarding quantum chemistry, quantum error correction would enable the use of the various Hamiltonian simulation algorithms, as described in Section 4.1.”
Quote in the paperSubstantial claim - Interpretationreasoning
With quantum error correction, the quantum phase estimation algorithm could become a viable alternative option to the variational quantum eigensolver for ground state energy estimation.
“Thus, with quantum error correction, the quantum phase estimation algorithm could become a viable alternative option to the variational quantum eigensolver for ground state energy estimation.”
Quote in the paperSubstantial claim - Interpretationreasoning
Insights from quantum chemistry have led to substantial improvements in quantum algorithms.
“Already, insights from quantum chemistry have led to substantial improvements in quantum algorithms.”
Quote in the paperNot a substantial claim - Interpretationreasoning
Given the setup, a hybrid classical-quantum sampling algorithm can efficiently generate the FCP profile at T = 0 K.
“Given this setup, the following hybrid classical-quantum sampling algorithm can efficiently generate the FCP profile at T = 0 K:”
Quote in the paperSubstantial claim - Interpretationreasoning
Experimental progress in adiabatic quantum computing for chemistry applications hinges on the development of non-gadgetized, highly tunable multi-spin couplers that either directly encode the problem Hamiltonian or reduce the mapping overhead of gadgetization.
“Therefore, experimental progress in the field of adiabatic quantum computing for chemistry applications hinges on the development of non-gadgetized, highly tunable multi-spin couplers that either directly encode the problem Hamiltonian or reduce the mapping overhead of gadgetization.”
Quote in the paperSubstantial claim - Premisecitation
Shen et al. 378 reproduce the molecular vibronic signal of SO2 using solely the vibrational excitations in a trapped ion device.
“In particular, Shen et al. 378 reproduce the molecular vibronic signal of SO 2 , using solely the vibrational excitations in a trapped ion device.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [378] McArdle, S.; Yuan, X.; Benjamin, S. Error Mitigated Digital Quantum Simulation. 2018, arXiv:1807.02467 [quant-ph]; https://arxiv.org/abs/1807.02467.
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- Interpretationreasoning
Quantum algorithm development also serves to influence the estimates of the quantum utility crossover.
“However, quantum algorithm development also serves to influence these estimates.”
Quote in the paperNot a substantial claim - Premisereasoning
The trade-off or cost of VQE is the large number of measurements needed to approximate the ground state energy with high precision.
“the trade-off or cost of the algorithm is the large number of measurements needed to approximate the ground state energy with high precision.”
Quote in the paperNot a substantial claim - Premisenone given
In practice, we can only obtain a finite number of measurements, leading to errors in the energy estimation.
“In practice, we can only obtain a finite number of measurements, leading to errors in the energy estimation.”
Quote in the paperSubstantial claim - Premisecitation
The Bravyi-Kitaev mapping combines advantages from the Jordan-Wigner and parity mappings to yield creation and annihilation operators which act non-trivially on O(log N) qubits.
“The Bravyi-Kitaev mapping 180 combines the advantages from the Jordan-Wigner and parity mappings to yield creation and annihilation operators which act non-trivially on O (log N ) qubits (i.e. have weight O (log N )).”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [180] source
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- Premisereasoning
In the Jordan-Wigner mapping, while each spin-orbital occupation operator is local, other one-body operators can be very non-local due to the high weight of the parity.
“So, while each spin-orbital occupation operator a † i a i is local, acting only on the i th qubit, other one-body operators a † i a j can be very non-local due to the high weight of the parity.”
Quote in the paperNot a substantial claim - Premisecitation
The dual PW basis set provides a more compact representation of the molecular Hamiltonian in second quantization.
“The dual PW basis set has the advantage of providing a more compact representation of the molecular Hamiltonian in second quantization.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [301] source
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- Premisecitation
The Bravyi-Kitaev tree method enables the use of qubit-reduction techniques when the spin-orbital number is not a power of two.
“However, it has been noted 358 that, in contrast to the standard Bravyi-Kitaev mapping, the Bravyi-Kitaev tree method enables the use of qubit-reduction techniques when the spin-orbital number is not a power of two.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [358] source
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- Premisenone given
Over the last five years, the asymptotic scaling of quantum simulation algorithms for quantum chemistry has been dramatically improved from a high-degree polynomial to sublinear in the number of orbitals.
“Over the last five years, as outlined in Section 4.2.2, the asymptotic scaling of quantum simulation algorithms for quantum chemistry have been dramatically improved from a high-degree polynomial to sublinear in the number of orbitals.”
Quote in the paperNot a substantial claim - Premisecitation
In the original Bravyi-Kitaev encoding for N = 2^n, the maximum weight of any encoded creation operator is exactly log2 N.
“In this case the maximum weight of any encoded a † j is exactly log 2 N according to Havl´ ıˇ cek et al. 402 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [402] source
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- Premisecitation
The estimated dates for reaching the quantum utility horizon are strongly determined by progress in hardware development and are extrapolated based on recent advances in quantum device technology.
“The estimated dates for reaching this horizon are strongly determined by progress in hardware development and are extrapolated based on recent advances in quantum device technology 56,385,396,397 .”
Quote in the paperNot a substantial claim0 of 4 sources checked - Interpretationown data
Provided that the ansatz can well describe the ground state and the classical optimizer is robust against noise in the cost function landscape, VQE can provide a high-quality estimation for the ground state energy.
“Provided that the ansatz can well describe the ground state and the classical optimizer is robust against noise in the cost function landscape, VQE can provide a high-quality estimation for the ground state energy.”
Quote in the paperSubstantial claim - Interpretationreasoning
Solutions to the particle-in-a-box problem can offer a better description of periodic systems, giving rise to plane-wave basis sets.
“Correspondingly, solutions to the particle-in-a-box problem can offer a better description of periodic systems, giving rise to plane-waves (PW) basis sets, which have a complex exponential form.”
Quote in the paperNot a substantial claim - Interpretationreasoning
The high weight of the parity-counting phase can be costly for certain quantum simulation algorithms.
“The high weight of the parity-counting phase can be costly for certain quantum simulation algorithms.”
Quote in the paperNot a substantial claim - Premisecitation
Symmetries in the four-qubit Hamiltonian for H2 can be exploited to construct a two-qubit Hamiltonian representing the symmetry sector of the original Hamiltonian that contains the ground state.
“As described in O'Malley et al. 55 , symmetries in this Hamiltonian can be exploited to construct a two-qubit Hamiltonian representing the symmetry sector of the original Hamiltonian that contains the ground state.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites O'Malley et al. 55
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- Premisereasoning
In the Jordan-Wigner mapping, creation operators act non-trivially on i qubits.
“Notice that the creation operators a † i act non-trivially on i qubits.”
Quote in the paperNot a substantial claim - Premisecitation
The choice and/or design of the ansatz largely influences the performance of VQE.
“Often the challenge in VQE is the choice and/or design of the ansatz, which largely influences the performance of the algorithm 321 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [321] source
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- Premisecitation
In practice, the Bravyi-Kitaev tree method produces higher-weight creation and annihilation operators than the standard Bravyi-Kitaev mapping.
“Although the scaling of operator weight is the same as that of the standard Bravyi-Kitaev mapping, in practice the Bravyi-Kitaev tree method produces higher-weight creation and annihilation operators 358 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [358] source
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- Premisereasoning
Energy bands in periodic systems are different from atomic orbitals.
“However, energy bands in periodic systems are different from atomic orbitals.”
Quote in the paperNot a substantial claim - Premisereasoning
The Born-Oppenheimer approximation is justified for the H2 problem because the ratio of electronic to nuclear mass is roughly 1:1000, leading to a separation in the timescale of their dynamics.
“This is justified as the ratio of electronic to nuclear mass is roughly 1:1000, leading to a separation in the timescale of their dynamics.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The Born-Oppenheimer approximation, treating the nuclei as stationary classical particles, is valid for the molecular hydrogen problem.
- Classified as probably not a substantial claim. 232
- May combine two claims. 135
- The quote only partly states this claim. 1
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PremisecitationThe first quantum algorithms suited particularly for quantum chemistry appeared as early as the late 1990s, including simulating fermionic Hamiltonians and calculating thermal rate constants efficiently.
“The first quantum algorithms suited particularly for quantum chemistry appeared as early as the late 1990s. These include, for instance, simulating fermionic Hamiltonians 43 and the quantum algorithm for calculating thermal rate constant efficiently 44 .”
Quote in the paperNot a substantial claim0 of 2 sources checkedProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Premisenone givenNot all quantum mechanical systems are difficult to simulate; some have easily computable exact solutions and others have clever computational shortcuts leading to approximate solutions.
“It is not true that all quantum mechanical systems are difficult to simulate; some of them have easily computable exact solutions and others have very clever computational shortcuts leading to approximate solutions.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationIn 1996, Lloyd provided a concrete method for using quantum computers to efficiently simulate the dynamics of other quantum systems.
“In 1996, Feynman's vision was manifested in a theoretical proposal by Lloyd 37 , providing a concrete method for using quantum computers to efficiently simulate the dynamics of other quantum systems.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [37] source
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PremisecitationThe opening decade of the 21st century witnessed the first quantum algorithms for quantum chemistry built on QPEA, for example algorithms for computing molecular spectra with exponential speedup over classical computers.
“The opening decade of the 21st century witnessed the first quantum algorithms for quantum chemistry built on the insights of QPEA. For example, the quantum algorithms developed for computing molecular spectra with exponential speedup over classical computers 45,46 .”
Quote in the paperNot a substantial claim0 of 2 sources checked- cites [45] source
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- cites [46] Manin Y., 1980
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PremisecitationSimulating quantum mechanics on classical computers requires resources growing exponentially in the problem instance size, so certain problems will remain out of reach regardless of algorithm design.
“Both argued that simulating quantum mechanics on classical computers requires resources growing exponentially in the problem instance size, so that certain problems will remain out of reach, regardless of the ingenuity of the algorithm designers.”
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PremisecitationWiesner and Zalka suggested using quantum computers for simulating the dynamics of many-body systems.
“Around the same time, Wiesner 40 and Zalka 41 also suggested using quantum computers for simulating the dynamics of many-body systems.”
Quote in the paperNot a substantial claim0 of 2 sources checkedMay combine two claims.
PremisereasoningExact diagonalization achieves the FCI answer in a single calculation without stochastic error bars, unlike QMC which arrives at it through extrapolation.
“While QMC arrives at this answer through extrapolation, ED achieves it in a single calculation by exact diagonalization of the Hamiltonian matrix without stochastic error bars.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationCoupled cluster theory is formulated as a compact wave function parametrization in terms of an exponential functional of a cluster operator.
“Coupled cluster 79 is formulated as a compact wave function parametrization in terms of an exponential functional of a cluster operator.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [79] source
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Premisenone givenA sign problem occurs in both diffusion Monte Carlo and auxiliary field QMC, but is mitigated by defining phase constraints on the auxiliary fields from a trial wave function.
“A sign problem occurs in both DMC and AFQMC, but is mitigated by defining phase constraints on the auxiliary fields from a trial wave function.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
PremisereasoningThe self-consistent field (SCF) approximation reduces the electron Hamiltonian to a mean-field potential determined self-consistently and describes the ground-state wave function with a single Slater determinant.
“Another way to avoid an exponential parameter scaling is by reducing the electron Hamiltonian to its self-consistent field (SCF) part. This approximates the system as being comprised of electrons experiencing a mean field potential, which has to be determined self-consistently.”
- Also stated in: § 2.1 Basics and challenges of classical quantum chemistry · p. 15 · plain
“The ground-state wave function is described using a single Slater determinant or configuration.”
Quote in the paperNot a substantial claim- Also stated in: § 2.1 Basics and challenges of classical quantum chemistry · p. 15 · plain
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PremisereasoningThere is no uniform answer on which DFT functional to use for which chemical systems.
“There have been several efforts to improve DFT functionals systematically (e.g. GGA, meta-GGA, hybrid approaches) but there is no uniform answer on which functional to use for which chemical systems.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationDensity functional theory (DFT) integrates out all electronic degrees of freedom except the density to avoid treating the full many-electron wave function.
“For systems of electrons, one common methodology to avoid treating the size of the full many-electron wave function is density functional theory (DFT) 77,78 . All degrees of freedom of the electronic system are integrated out, except for the density.”
Quote in the paperNot a substantial claim0 of 2 sources checkedProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationUnder the assumption of a gap between the ground state and first excited state during adiabatic evolution, the end of the adiabatic evolution should be close to the ground state of H1.
“Assuming there is always a gap between the ground state of H ( s ) and the first excited state, the end of the adiabatic evolution of the state of the system should be close to the ground state of H 1 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [60] Preskill, J. Quantum Computing in the NISQ Era and Beyond. 2018, arXiv:1801.00862 [quant-ph]; https://arxiv.org/abs/1801.00862. source
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- The authors state this as an assumption: There is always a gap between the ground state of H(s) and the first excited state during the adiabatic evolution.
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PremisecitationExact diagonalization methods provide the exact wave function and energy within a certain basis set (the FCI answer).
“The exact diagonalization (ED) methods 79,92,93 provide the exact answer for the wave function and energy within a certain basis set (i.e. the Full Configuration Interaction (FCI) answer).”
Quote in the paperNot a substantial claim0 of 3 sources checked- cites [79] source
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- cites [92] Helgaker T., 2000 source
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- cites [93] Fan, P.D.; Piecuch, P. In The Usefulness of Exponential Wave Function Expansions Employing One- And Two-Body Cluster Operators in Electronic Structure Theory: The Extended and Generalized Coupled-Cluster Methods ; Sabin, J., Brändas, E., Eds. Advances in Quantum Chemistry; Academic Press, 2006; Vol. 51, pp 1 – 57. source
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PremisecitationThe Boson Sampling model does not have the property of computational equivalence to the gate model.
“In contrast, the Boson Sampling model of computation does not have this property 63 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [63] source
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PremisereasoningThis intractability has inspired formulations of various physically-motivated wave function approximations and quantum chemistry algorithms on classical computers.
“This has inspired formulations of various physically-motivated wave function approximations as well as quantum chemistry algorithms on classical computers.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationThe AQC model is computationally equivalent to the gate model, meaning algorithms written in either model can be translated to the other without prohibitive computational overhead.
“The AQC model has been shown to be computationally equivalent to the gate model 62 , meaning that algorithms written in either model can be translated to ones in the other model without incurring prohibitive computational overhead.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [62] source
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Premisenone givenThe quality of DFT depends on the chosen exchange-correlation functional.
“However, the quality of DFT depends on the chosen exchange-correlation functional.”
Quote in the paperNot a substantial claimMay combine two claims.
InterpretationcitationQuantum computers avoid the dynamical sign problem, providing a more effective alternative to classical methods for quantum simulation.
“It has been shown that quantum computers avoid the dynamical sign problem 43 providing a more effective alternative to classical methods for quantum simulation.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [43] source
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Premisenone givenMost DFT functionals give reasonable results around equilibrium geometries but behave unpredictably in the regime of strong correlations or in the presence of dispersive electron interactions, such as bond breaking or solvation chemistry.
“Most functionals tend to give reasonable results around equilibrium geometries but behave unpredictably in the regime of strong correlations or in the presence of dispersive electron interactions, e.g. bond breaking or solvation chemistry.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationThe Boson Sampling model gives rise to a class of sampling problems that are hard to solve on a classical computer.
“However, it does give rise to a class of sampling problems 64 which are hard to solve on a classical computer.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [64] source
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PremisecitationA diverse set of experimental groups worldwide have already implemented various variational quantum algorithms on their physical systems.
“A diverse set of experimental groups worldwide have already implemented various variational quantum algorithms on their physical systems 52-59 .”
Quote in the paperNot a substantial claim0 of 8 sources checked- cites [52] Wiesner, S. Simulations of Many-Body Quantum Systems by a Quantum Computer. 1996, arXiv:quant-ph/9603028; https://arxiv.org/abs/quant-ph/9603028.
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- cites [53] source
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- cites [54] Kitaev A., 2002 source
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- cites [55] source
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- cites [56] source
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- cites [57] Hastings M. B., 2015, Quantum Inf. Comput.
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- cites [58] Poulin D., 2014, Quantum Inf. Comput.
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- cites [59] source
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Premisenone givenQMC relies on stochastic estimation of the energy of a trial wave function.
“QMC relies on the stochastic estimation of the energy of a trial wave function”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.
Premisenone givenFixed-node diffusion Monte Carlo biases the population but is more robust and provides accurate results.
“A popular implementation of DMC fixes the nodes of the trial wave function (fixed-node DMC) which biases the population but is shown to be more robust and provides accurate results.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationExact simulation of quantum chemistry systems is widely regarded as one of the problems that would benefit enormously from quantum hardware.
“Exact simulation of quantum chemistry systems is widely regarded as one of the problems that would benefit enormously from quantum hardware 29 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [29] source
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InterpretationreasoningSCF completely neglects correlation effects, so computational approximations for various physical quantities are usually of low quality.
“Correlation effects are thereby completely neglected, meaning computational approximations for various physical quantities are usually of low quality.”
Quote in the paperSubstantial claimMay combine two claims.
PremisecitationVariationally optimized coupled cluster overcomes the non-variational problem but can only be applied to small systems or models with a reduced parameter space on a classical computer.
“Variationally optimized coupled cluster 84 overcomes this problem but can only be applied to small systems 85-87 or models with a reduced parameter space 88 on a classical computer.”
Quote in the paperSubstantial claim0 of 5 sources checkedProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Premisenone givenQuantum Monte Carlo (QMC) has been very successful in performing large-scale calculations for extended systems in quantum chemistry.
“It has been very successful in performing large-scale calculations for extended systems in quantum chemistry.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S4 (based). May combine two claims.
Premisenone givenIn variational quantum algorithms, the quantum computer only needs to perform a short sequence of operations, unlike QPEA-based algorithms which require a long sequence of operations.
“Unlike the algorithms based on QPEA, which require the quantum computer to perform a long sequence of operations, in the variational quantum algorithm framework the quantum computer only needs to perform a short sequence of operations.”
Quote in the paperNot a substantial claimClassified as probably not a substantial claim.
Premisenone givenThe exact solution for an electronic system with a fixed basis set is a full configuration interaction (FCI) expansion, a variationally determined superposition of all determinants in the N-particle Fock space.
“The exact solution to Equation 1 for an electronic system described with a fixed set of basis set functions consists of a variationally determined superposition of all determinants in the N -particle Fock space, also known as a full configuration interaction (FCI) expansion.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationThe quantum computational cost of quantum chemistry algorithms has continued to be reduced.
“The quantum computational cost of these algorithms has continued to be reduced 47-50 .”
Quote in the paperNot a substantial claim0 of 4 sources checkedProbably not a substantial claim: may not pass S3 (load-bearing).
InterpretationreasoningFCI is still most useful as a benchmarking method for less-costly quantum chemical methods.
“Despite all these advances, FCI is still the most useful as a benchmarking method for less-costly quantum chemical methods.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).
InterpretationreasoningThese complexity characterizations provide further motivation for developing quantum algorithms targeted at solving them.
“These characterizations, discussed in Chapter 3, provide further motivation for the development of quantum algorithms targeted at solving them.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.
Premisenone givenThe coefficients h_pq and h_pqrs are one-electron and two-electron integrals which can be efficiently computed classically for many choices of basis functions.
“Here the coefficients h pq and h pqrs are one-electron and two-electron integrals which can be efficiently computed classically for many choices of basis functions.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationClassical molecular dynamics (MD) simulates large chemical systems using an effective force field and Newtonian mechanics, avoiding explicit wave function propagation.
“It is sufficient to use classical MD methods which simulate the problem using an effective force field and Newtonian mechanics 140 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [140] source
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Premisenone givenNISQ devices are available at present or use technologies available in the near-term, while fault-tolerant quantum computing devices require much more research and are a much longer-term prospect.
“The first are available at present or are using technologies that will be available in the near-term, while the latter require much more research and are a much longer-term prospect.”
Quote in the paperNot a substantial claimClassified as probably not a substantial claim.
Premisenone givenA long line of inquiry in quantum computing has led to a refined set of quantum algorithms for efficiently simulating time evolution under both time-independent and time-dependent Hamiltonians.
“For the latter feature, time evolution, a long line of inquiry in quantum computing that spans more than two decades has led to a refined set of quantum algorithms for efficiently simulating time evolution under both time-independent and time-dependent Hamiltonians.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Premisenone givenConical intersections occur when two electronic states intersect at a point in nuclear coordinate space; close to these points the Born-Oppenheimer approximation breaks down, resulting in entanglement between electronic and nuclear degrees of freedom.
“Conical intersections occur when two electronic states, e.g. the ground state and first excited state of a molecule, intersect at a point in nuclear coordinate space. Close to these points, the Born-Oppenheimer approximation breaks down, resulting in entanglement between electronic and nuclear degrees of freedom.”
Quote in the paperNot a substantial claim- The authors state this as an assumption: The Born-Oppenheimer approximation treats nuclei as stationary point charges because nuclear mass is three orders of magnitude greater than electron mass.
May combine two claims.
PremisereasoningExact time evolution requires knowledge of the full spectrum of the Hamiltonian and the expansion of the initial state in eigenvectors, which is only feasible for very small systems.
“exact time evolution requires knowledge of the full spectrum of the Hamiltonian and the expansion of the initial state in terms of the eigenvectors. This is only feasible for very small systems.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
PremisecitationDMRG variationally optimizes a wave function in the form of a matrix product state (MPS), a one-dimensional chain of tensors.
“Specifically, DMRG variationally optimizes a wave function in the form of a matrix product state (MPS) 110 :”
- Also stated in: § 2.2.1 Static properties: wave functions and energy spectra · p. 19 · plain
“which is a one-dimensional chain of tensors.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [110] source
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- Also stated in: § 2.2.1 Static properties: wave functions and energy spectra · p. 19 · plain
May combine two claims.
PremisecitationMatrix product states allow for a compact representation of one-dimensional systems and satisfy an area law in one dimension.
“Matrix product states allow for a compact representation of one-dimensional systems and also satisfy an area law 111 in one dimension.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [111] Harrison R., 1994
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PremisecitationNuclear quantum effects are needed for simulating condensed matter systems containing low-temperature helium and hydrogen, and for high-accuracy simulations of hydrogen-containing room-temperature systems such as liquid water.
“Nuclear quantum effects are needed for simulating condensed matter systems containing low-temperature helium and hydrogen 143 , as well as for obtaining high accuracy simulations of hydrogen-containing room-temperature systems such as liquid water 144 .”
Quote in the paperNot a substantial claim0 of 2 sources checkedMay combine two claims.
PremisereasoningPath integral methods model nuclear quantum effects by discretizing an atom into many beads, avoiding a Hilbert space and limiting computational cost to that of classical MD.
“Conceptually, these methods operate by discretizing an atom into many 'beads' that represent a state of the system in the path-integral. While these methods include some quantum effects listed above, they avoid using a Hilbert space, which limits their computational cost to that of classical MD methods.”
Quote in the paperNot a substantial claim0 of 3 sources checkedProbably not a substantial claim: may not pass S4 (based).
Premisenone givenFor strongly correlated (multireference) systems where a single dominating configuration breaks down, traditional single-reference electronic structure methods are usually not applicable.
“In the case of strongly correlated (multireference) systems where the concept of a single dominating configuration breaks down, traditional single reference electronic structure methods are usually not applicable.”
Quote in the paperNot a substantial claimMay combine two claims.
PremisecitationMCTDH models all relevant quantum degrees of freedom, propagating superpositions of product states, but evolution over longer time periods requires considerable computer resources, so larger systems are out of scope of current capabilities.
“The multiconfiguration timedependent Hartree (MCTDH) algorithm 175 models all relevant quantum degrees of freedom, propagating superpositions of product states. Evolution of a wave packet over longer time periods in MCTDH requires considerable computer resources and therefore larger systems are out of the scope of current capabilities.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [175] source
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PremisecitationEarly proposals for representing wave functions of quantum systems consider a real-space representation where the spatial wave function is discretized into grid points.
“Early proposals 40,41 for representing wave functions of quantum systems consider a real-space representation where the spatial wave function ψ ( x ) is discretized into grid points ∑ i ψ i | i 〉 with each | i 〉 being a computational basis state.”
Quote in the paperNot a substantial claim0 of 2 sources checkedMay combine two claims.
PremisecitationKrylov subspace methods make it possible to approximate e^{-iH}v for a given vector v by solely performing matrix-vector multiplications, which is efficient for large matrices.
“Finally, the Krylov subspace methods (e.g. the Lanczos method for Hermitian matrices) make it possible to approximate e -iH v for a given vector v , by solely performing matrix-vector multiplications 154,157 . This is efficient for large matrices.”
Quote in the paperNot a substantial claim0 of 2 sources checkedProbably not a substantial claim: may not pass S3 (load-bearing).
Premisenone givenThe main difference between NISQ and FTQC devices concerns decoherence due to environmental noise and whether it sets an upper limit on the time duration of a quantum computation.
“The main difference between NISQ and FTQC devices concerns decoherence due to the environmental noise and whether it sets an upper limit on the time duration of a quantum computation.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
PremisecitationStandard ODE solvers such as Runge-Kutta can be adapted to quantum propagation but are not the most computationally efficient choice for a given accuracy.
“They can be adapted to perform quantum propagation 150,152 , although they are not the most computationally efficient choice for a given accuracy.”
Quote in the paperNot a substantial claim0 of 2 sources checkedProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationThe time-dependent variational principle has enabled time-dependent variational Monte Carlo calculations for bosons and strongly correlated fermions.
“The time-dependent variational principle (TDVP) 167-169 has enabled time-dependent variational Monte Carlo calculations for bosons 170 and strongly correlated fermions 171 .”
Quote in the paperNot a substantial claim0 of 5 sources checkedMay combine two claims.
PremisecitationPolynomial methods rely on exact expression of high-order matrix powers as lower-order polynomials of the Hamiltonian, with the bottleneck being calculation of the characteristic polynomial, which is expensive for large matrices.
“They rely on the exact expression of high-order matrix powers as lower-order polynomials of the Hamiltonian 153 . The bottleneck for those methods is calculating the characteristic polynomial, which is expensive for large matrices.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [153] Leach A., 2001
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Premisenone givenNuclear quantum effects such as nuclear tunneling and zero-point vibrational energy are not captured by Newtonian mechanics.
“Such effects include nuclear tunneling and zero-point vibrational energy, which are not captured by Newtonian mechanics.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing), S4 (based).
Interpretationnone givenThere are promising ideas which may bring tangible results for quantum chemistry on near-term devices.
“In fact, there are promising ideas which may bring tangible results for quantum chemistry on near-term devices.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Premisenone givenMD is often applied to problems such as diffusion rates, reaction rates, conformal changes, infrared spectra, and binding energies, and to processes such as protein folding, gas absorption in porous materials, stress-induced material deformation, and electrolyte solvation.
“MD is often applied to problems such as diffusion rates, reaction rates, conformal changes, infrared spectra, and binding energies, to study processes such as protein folding, gas absorption in porous materials, stress-induced material deformation, and electrolyte solvation, among many others.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S4 (based).
Premisenone givenThere are quantum states for which no known efficient classical representation exists, but that can be prepared on a quantum computer.
“For the former, i.e. state representation, there are quantum states for which no known efficient classical representation exists, but that can be prepared on a quantum computer.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Premisenone givenMany well-known quantum algorithms such as Shor's factorization, Grover's search, and quantum phase estimation assume the availability of FTQC devices to deliver promised quantum advantage.
“Many well-known quantum algorithms that have become standard examples of quantum speedups over classical algorithms, such as Shor's quantum factorization algorithm, quantum search algorithm by Grover and quantum phase estimation (which we will review here), in fact assume the availability of FTQC devices in order to deliver the promised quantum advantage.”
Quote in the paperNot a substantial claimMay combine two claims.
InterpretationreasoningThe full quantum state near conical intersections is no longer well-approximated with a product state, necessitating methods that allow for multireference character in the wave function.
“This means that the the full quantum state is no longer well-approximated with a product state, necessitating methods that allow for multireference character in the wave function, of which we mention three popular methods here.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
PremisereasoningNISQ devices are sensitive to sources of noise and error, while FTQC devices can, in principle, carry out coherent quantum computation for an arbitrary amount of time.
“In this sense, NISQ devices are sensitive to sources of noise and error (hence the term 'noisy' in NISQ) while FTQC devices can, in principle, carry out coherent quantum computation for an arbitrary amount of time.”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationnone givenQuantum computers allow for efficient implementation of explicitly maintaining the full many-body wave function and propagating the wave function in time by general matrix-vector multiplication.
“However, as we will discuss in this section, quantum computers allow for efficient implementation of these two features.”
Quote in the paperSubstantial claimMay combine two claims.
InterpretationreasoningThe long-range Coulomb interaction in molecules causes entanglement among molecular orbitals to be multidimensional, so larger bond dimension MPS matrices must be employed for generic non-one-dimensional molecules to capture correlations.
“However, the long range Coulomb interaction in molecules causes the entanglement among the MOs to be multidimensional as opposed to one-dimensional. As a result, larger dimension MPS matrices (or 'high bond dimension' MPS) have to be employed in case of generic (non-one-dimensional) molecules in order to properly capture correlations among MOs.”
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Premisenone givenThe second-quantized picture, where each computational basis state corresponds to a Slater determinant in Fock space, underlies the majority of quantum algorithms for quantum chemistry on both NISQ and FTQC devices.
“An alternative representation considers the second-quantized picture where each computational basis state | i 1 i 2 · · · i n 〉 corresponds to a Slater determinant in the Fock space. Each i j = 1 if the spin-orbital j is occupied and 0 otherwise. This representation underlies the majority of quantum algorithms for quantum chemistry on both NISQ and FTQC devices.”
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Premisenone givenLinear response is the first-order expansion of a time-dependent property and perturbation theory falls short of describing the full time dependence.
“Linear response is the first-order expansion of a time-dependent property and perturbation theory falls short of describing the full time dependence.”
Quote in the paperNot a substantial claimMay combine two claims.
Premisenone givenMatrix decomposition methods approximate the operator exponential directly and require far fewer matrix-matrix operations than the naive implementation.
“Matrix decomposition methods try to approximate the operator exponential of the Hamiltonian directly. The prototypical example of this is the decomposition of the exponential as e A = Se V S -1 , where A = SV S -1 and V is diagonal. This approach requires far fewer matrixmatrix operations than the naive implementation.”
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Premisenone givenZero-temperature ground-state quantum chemistry covers only a restricted set of chemistry in nature; many processes occur at finite temperature and are described by the time-dependent Schrödinger equation.
“Zero-temperature ground-state quantum chemistry covers only a restricted set of the chemistry in nature. Many processes occur at finite temperature where the system is propagated in time and described by the time-dependent Schr¨ odinger equation”
Quote in the paperNot a substantial claimMay combine two claims.
PremisecitationDirectly propagating the solution of the Schrödinger equation by updating an explicit wave function or density matrix at each time step quickly becomes memory intensive and requires approximating the matrix exponential of the Hamiltonian.
“One of the strategies for time propagation is directly propagating the solution of the Schr¨ odinger equation, i.e. updating an explicit wave function or density matrix at each time step. This quickly becomes memory intensive and requires a method to approximate the matrix exponential of the Hamiltonian 150 .”
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InterpretationcitationA problem being NP-complete is strong evidence that there does not exist a provably efficient algorithm to solve it, whether classical or quantum.
“Roughly speaking, a problem being NP -complete is strong evidence that there does not exist a provably efficient algorithm to solve it, be it of classical or quantum nature 204 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [204] Low, G. H.; Chuang, I. L. Hamiltonian Simulation by Qubitization. 2016, arXiv:1610.06546 [quant-ph]; https://arxiv.org/abs/1610.06546.
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Premisenone givenAny discussion of realizing e^{-iHt} is restricted to simulating time evolution of closed, time-independent systems.
“Naturally, any discussion of realizing the operation e -iHt is restricted to simulating time evolution of closed , time-independent systems.”
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PremisereasoningBy iterating between classical and quantum processors, a hybrid algorithm eventually trains the quantum computer to prepare the desired state for the problem.
“By iterating between classical and quantum processors, the hybrid algorithm eventually trains the quantum computer to prepare the state that is desired for the problem.”
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PremisereasoningA salient feature of hybrid quantum-classical algorithms is that much of the computational burden is offloaded to a classical computer.
“A salient feature of hybrid quantum-classical algorithms is that much of the computational burden is offloaded to a classical computer.”
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Premisenone givenThe variational quantum eigensolver (VQE) is an important class of hybrid quantum-classical algorithms useful for solving static problems in quantum chemistry.
“An important class of hybrid quantum-classical algorithms that are useful for solving static problems in quantum chemistry is the variational quantum eigensolver (VQE)”
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PremisecitationShor's algorithm for integer factorization and amplitude estimation can both be cast as forms of phase estimation for specific constructions of unitary operators U.
“For example, Shor's algorithm for integer factorization 32 and amplitude estimation 183 both can be cast as a form of phase estimation for specific construction of unitary operators U .”
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Premisenone givenHamiltonians for molecular electronic structure problems are well studied in the context of implementing time evolution U = e^{-iHt}.
“In addition, the Hamiltonians for molecular electronic structure problems are also well studied in the context of implementing time evolution U = e -iHt .”
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Premisenone givenPhase estimation is an algorithmic framework that encapsulates a broad set of quantum algorithms.
“Phase estimation is an algorithmic framework that encapsulates a broad set of quantum algorithms.”
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PremisereasoningThe time evolution of an initial state translates to updating the classical parameters iteratively as the simulation progresses.
“The time evolution of an initial state | ψ ( ⃗ θ 0 ) 〉 then translates to updating the classical parameters iteratively as the simulation progresses.”
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PremisecitationIt is unknown whether efficient checking of a candidate solution implies efficient solving; this is the unsolved P vs NP conjecture.
“It is unknown whether or not a problem being efficient to check implies that it is efficient to solve. This question is captured by the famous P vs NP conjecture 203 that is still unsolved.”
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InterpretationcitationQuantum computation circumvents the main issues encountered in classical methods by maintaining a highly entangled quantum state that may not admit an efficient classical description, enabling a fundamentally different approach to realizing unitary evolution and avoiding the dynamical sign problem.
“Quantum computation circumvents the main issues encountered in classical methods by maintaining a highly entangled quantum state which may not admit efficient classical description, enabling a fundamentally different approach to realizing the unitary evolution and in doing so avoids the dynamical sign problem 43 .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [43] source
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PremisecitationFor time-dependent Hamiltonians, early works focus on variants of the Trotter-Suzuki formula, which are subsequently improved with black-box models.
“For timedependent Hamiltonians, early works 196 focus on variants of Trotter-Suzuki formula, which are subsequently improved with black-box models 188 .”
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Premisenone givenThere are many examples of elusive problems in quantum physics or quantum chemistry that seem to defy efficient solutions despite decades of effort, including the 3D Ising problem, Hubbard models, and finding the universal functional in density functional theory.
“There are many examples of elusive problems in quantum physics or quantum chemistry that seem to defy efficient solutions despite decades of effort. These include the 3D Ising problem, Hubbard models, and finding the universal functional in density functional theory.”
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PremisereasoningClassical simulation of time evolution is tractable when the time step t is small, yielding a linear response theory.
“the time step t is small, for which e -iHt ≈ I -iHt . This yields a linear response theory;”
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PremisereasoningBy measuring the state |~φ_j> one could extract the phase φ_i up to the precision allowed by the qubit register.
“By measuring the state | ˜ ϕ j 〉 one could extract the phase ϕ i up to the precision allowed by the qubit register.”
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Premisenone givenAlthough the exact relationships between complexity classes remain largely open, certain containment relationships have already been proven.
“Although the exact relationships between the complexity classes remain largely open, certain containment relationships have already been proven (Figure 2).”
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PremisereasoningThe computational complexity of a given problem refers to its worst-case hardness and does not always inform how difficult the problem is to solve in practice.
“It is important to note that the computational complexity of a given problem refers to its worst-case hardness, and does not always inform us of how difficult a problem is to solve in practice.”
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PremisecitationAlgorithms for simulating open quantum systems have been explored; they focus on the Hamiltonian and Lindblad operators being sparse and given by oracles.
“More generally, algorithms for simulating open quantum systems have also been explored 193-195 . These algorithms focus on the Hamiltonian and Lindblad operators being sparse and given by oracles (black-boxes).”
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- cites [194] source
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- cites [195] von Neumann J., 1949
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- The authors state this as an assumption: In algorithms for simulating open quantum systems, the Hamiltonian and Lindblad operators are sparse and given by oracles or black boxes.
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PremisereasoningProblems in #P are at least as hard as the corresponding problems in NP.
“There is a sense in which problems in #P are at least as hard as the corresponding problems in NP .”
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InterpretationreasoningIf a problem is QMA-complete, then it is unlikely that there is an efficient solution even on a quantum computer.
“Drawing a parallel from the discussion above, if a problem is QMA -complete then it is unlikely that there is an efficient solution even on a quantum computer.”
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Premisenone givenFor quantum chemistry applications, there are well motivated methods for preparing initial wave functions based on efficiently computable ansatzes.
“For quantum chemistry applications, there are well motivated methods for preparing initial wave functions based on efficiently computable ansatzes.”
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PremisecitationSimulation techniques in the interaction picture have been proposed.
“Recently, simulation techniques in the interaction picture 197 have been proposed.”
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InterpretationreasoningPhase estimation circuits require fault-tolerant quantum computers, which can in principle perform quantum computation indefinitely.
“Instead they require fault-tolerant quantum computers which can in principle perform quantum computation indefinitely.”
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Premisenone givenPhase estimation can be used for extracting the eigenspectrum of a Hamiltonian H, corresponding to the special case U = e^{-iHt}.
“Most relevant to this review, is the use of phase estimation for extracting the eigenspectrum of a Hamiltonian H . This corresponds to the special case where U = e -iHt .”
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Premisenone givenThere are heuristic methods that regularly solve the Hartree-Fock problem in practice.
“Yet, there are heuristic methods which regularly solve the Hartree-Fock problem in practice.”
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Premisenone givenIn hybrid quantum-classical algorithms, the quantum computer is used only for preparing entangled states that are otherwise hard to represent classically and for making measurements with respect to those states.
“The quantum computer is only used for its ability to prepare entangled states that are otherwise hard to represent classically and to make measurements with respect to the states.”
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Premisecitation#BQP has been defined and shown to contain important physical problems, but it is of equal complexity to #P.
“While the quantum analog of #P , #BQP , has been defined and has been shown to contain important physical problems, it turns out to be of equal complexity to #P 67 .”
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Premisenone givenIn NISQ variational methods, a quantum state |ψ(θ)〉 is generated by a low-depth circuit with classical parameters θ.
“In these methods, a quantum state | ψ ( ⃗ θ ) 〉 is generated by a low depth circuit with classical parameters ⃗ θ .”
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Premisenone givenDetermining the optimal Hartree-Fock approximation of a molecular eigenstate is known to have the same complexity (NP-completeness) as many infamously hard problems.
“A pertinent example is that of determining the optimal Hartree-Fock approximation of a molecular eigenstate. This problem is known to have the same complexity ( NP -completeness) as many infamously hard problems.”
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Premisenone givenIn quantum chemistry, the standard task is to determine the ground state energy of a quantum system rather than directly simulate the dynamics of a quantum wave function.
“In quantum chemistry, it is relatively uncommon to directly simulate the dynamics of a quantum wave function. Instead, the standard task is to determine the ground state energy of a quantum system.”
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Premisenone givenIn practice, quantum computers are error-prone due to imprecision of processes and unwanted interaction with the environment.
“In practice, quantum computers are error-prone due to both imprecision of processes and unwanted interaction with the environment.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim. May combine two claims.
PremisecitationIt is possible to correct errors, and the system of detecting and correcting errors is called an error correcting code.
“However, it is possible to correct the errors and the system of detecting and correcting for errors is called an error correcting code 237 .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [237] Ryser H. J., 1963 source
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PremisecitationFinding the ground state of the Bose-Hubbard model is QMA-complete.
“finding the ground state of Bose-Hubbard model is QMA -complete 210”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [210] Cleve, R.; Wang, C. Efficient Quantum Algorithms for Simulating Lindblad Evolution. 2016, arXiv:1612.09512 [quant-ph]; https://arxiv.org/abs/1612.09512.
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PremisecitationThe original Aharonov and Ta-Shma work established a query complexity of O(poly(N,d)(τ^2/ε)), which matches the gate complexity of Lloyd's algorithm but applies to a broader class of d-sparse Hamiltonians.
“The original Aharonov and Ta-Shma work established a query complexity of O (poly( N,d )( τ 2 /ϵ )), which matches the gate complexity of Lloyd's algorithm, but applies to a broader class of d -sparse Hamiltonians 250 .”
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PremisecitationA rough, conservative estimate for the ratio of physical to logical qubits needed for reliable quantum computation is 10,000-to-1, but it depends on the realization of physical qubits, type of noise, and properties of the error correcting code.
“A rough, conservative estimate for the ratio of physical to logical qubits needed for reliable quantum computation is 10,000-to-1 237 but depends on the realization of physical qubits, type of noise and the properties of the error correcting code.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [237] Ryser H. J., 1963 source
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PremisecitationThe number of classical bits needed to store a wave function of N quantum systems grows as O(exp(N)), whereas the number of quantum bits needed to store this wave function grows as O(N).
“while the scaling of the number of classical bits needed to store a wave function of N quantum systems grows as O (exp( N )), the scaling of quantum bits needed to store this wave function is 1 O ( N ).”
Quote in the paperSubstantial claim0 of 2 sources checkedMay combine two claims.
InterpretationreasoningAlthough it is unlikely that one can find general and provably effective algorithms for QMA-complete or NP-complete problems, in practice there are heuristic methods that have gained some empirical success.
“Although it is unlikely that one can find general and provably effective (quantum or classical) algorithms for QMA -complete or NP -complete problems, in practice, there are heuristic methods that have gained some empirical success.”
Quote in the paperNot a substantial claimMay combine two claims.
PremisecitationThe sparse Hamiltonian approach provides a means for efficiently simulating Hamiltonians with fewer than d entries in each column (d-sparse) with gate count poly(N,t,1/ε,d), and includes local Hamiltonians as a special case.
“This approach provides a means for efficiently simulating (i.e. with gate count poly( N,t, 1 /ϵ, d )) Hamiltonians which have fewer than d entries in each column ( d -sparse) and includes local Hamiltonians as a special case.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [253] source
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PremisecitationLloyd's result applies to gate-based or digital quantum simulation, which is distinct from analog quantum simulation.
“We emphasize that Lloyd's result applies to a particular approach to quantum simulation, known as gate-based or digital quantum simulation . This approach is distinct from an analog quantum simulation , for which there have been several recent ground-breaking experiments 245-249 .”
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- cites [245] source
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- cites [246] source
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- cites [247] source
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- cites [248] Kitaev, A. Y. Quantum Measurements and the Abelian Stabilizer Problem. 1995, arXiv:quant-ph/9511026; https://arxiv.org/abs/quant-ph/9511026.
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- cites [249] source
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PremisecitationThe 3D Ising model is NP-complete.
“the 3D Ising model is NP -complete 209”
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PremisecitationLow and Chuang introduced the signal oracle U_signal such that in a subspace flagged by an ancilla in a signal state |G>, H = <G|U_signal|G>.
“Later, Low and Chuang 189 introduced the signal oracle U signal such that in a subspace flagged by an ancilla in a signal state | G 〉 , H = 〈 G | U signal | G 〉 .”
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PremisecitationFinding the universal functional in density functional theory is QMA-complete.
“finding the universal functional in density functional theory is also QMA -complete 65 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [65] Colless, J.; Ramasesh, V.; Dahlen, D.; Blok, M.; McClean, J.; Carter, J.; de Jong, W. A.; Siddiqi, I. Implementing a Variational Quantum Eigensolver Using Superconducting Qubits. Quantum Information and Measurement 2017 , 2017; p QF6A.2. source
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PremisecitationLloyd's analysis assumes that the Hamiltonian describes only local interactions.
“The structure assumed in Lloyd's analysis is that the Hamiltonian describes only local interactions .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [37] source
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Premisenone givenIn the case of Hartree-Fock, the self-consistent field iteration serves as a local search algorithm that can converge to a local optimum.
“For example in the case of Hartree-Fock, the self-consistent field iteration serves as a local search algorithm which can converge to a local optimum.”
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Premisenone givenThe advantage of the digital simulation approach is its aim for universality: any feasible Hamiltonian can be digitally simulated.
“The advantage of the digital simulation approach is its aim for universality : any feasible Hamiltonian can be digitally simulated.”
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Premisenone givenThe quantum algorithms discussed in Section 4 were designed for quantum computers that function without error.
“The quantum algorithms that we discuss in this section were designed for quantum computers that function without error.”
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PremisecitationThe principle behind quantum error correction is to use more physical qubits to establish a set of fewer, but higher-fidelity, equivalent qubits called logical qubits.
“In brief, the principle behind quantum error correction is to use more physical qubits to establish a set of fewer, but higher-fidelity, equivalent qubits called logical qubits 238 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [238] Gard B. T., 2015 source
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PremisereasoningThe algorithms described in Section 4 are typically beyond the limit of current error-prone quantum computers, so only proof-of-principle demonstrations for simple problems have been executed thus far.
“The algorithms described in this section are typically beyond this limit. Thus, only proof-of-principle demonstrations for simple problems have been executed thus far 55,234-236 .”
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PremisecitationAharonov and Ta-Shma introduced the sparse Hamiltonian paradigm and addressed the first two questions about non-local Hamiltonians and improved scaling.
“Independently, Aharonov and Ta-Shma 253 introduced the sparse Hamiltonian paradigm and thus addressed the first two questions above.”
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Premisenone givenState-of-the-art quantum algorithms achieve scaling roughly logarithmic in inverse precision and linear in time.
“The state-of-art quantum algorithms achieve scaling roughly logarithmic in inverse precision and linear in time.”
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Premisenone givenBy chance, the self-consistent field iteration can also reach the global optimum, but proving that it is the global optimum or systematically finding the global minimum is NP-complete.
“By chance, this can also be the global optimum, but proving that it is, or systematically finding the global minimum, is NP -complete.”
Quote in the paperNot a substantial claimClassified as probably not a substantial claim.
Premisenone givenThe exponential complexity in handling the exact wave function is well known to be a major obstacle in solving many important problems in quantum chemistry such as electronic structure calculations.
“The exponential complexity in handling the exact wave function is well known to be a major obstacle in solving many important problems in quantum chemistry such as electronic structure calculations.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Premisenone givenQuantum computers are particularly useful when classical methods cannot provide answers with high accuracy, such as for strongly correlated systems.
“Quantum computers are particularly useful when classical methods cannot provide answers with high-accuracy such as for strongly correlated systems.”
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Premisenone givenThe ansatz for state preparation comes from a classical approximation for a ground state of a given Hamiltonian.
“The ansatz for the preparation comes from a classical approximation for a ground state of a given Hamiltonian.”
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PremisecitationState preparation is conceptually easier in second quantization, where a Fock state is represented by a simple product state of qubits in {|0>, |1>} states.
“They pointed out that state preparation is conceptually easier in second quantization, where a Fock state is represented by a simple product state of qubits in {| 0 〉 , | 1 〉} states.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [38] Grover, L. K. A Fast Quantum Mechanical Algorithm for Database Search. Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing ; New York, 1996; pp 212–219. source
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PremisecitationThe quantum split-operator method is analogous to classical exact wave function simulation techniques such as matching pursuit/split-operator Fourier transform.
“The quantum 'split-operator' method 75,273 is analogous to classical exact wave function simulation techniques such as matching pursuit/split-operator Fourier transform 176 .”
Quote in the paperNot a substantial claim0 of 3 sources checked- cites [75] Aharonov, D.; van Dam, W.; Kempe, J.; Landau, Z.; Lloyd, S.; Regev, O. Adiabatic Quantum Computation Is Equivalent to Standard Quantum Computation. 45th Annual IEEE Symposium on Foundations of Computer Science , 2004.
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- cites [273] Childs, A. M.; Kothari, R. Simulating Sparse Hamiltonians With Star Decompositions, Conference on Quantum Computation, Communication, and Cryptography , 2010; pp 94–103. source
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- cites [176] source
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PremisecitationFor small molecules, HF provides a good approximation for the ground state, for example, 0.77 for FeMoCo.
“For small molecules, HF provides a good approximation for the ground state, for example, 0.77 for FeMoCo 287 .”
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PremisecitationAbrams and Lloyd described concrete initialization procedures for state preparation.
“Abrams and Lloyd 38 focused on state preparation in more detail and described concrete initialization procedures.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [38] Grover, L. K. A Fast Quantum Mechanical Algorithm for Database Search. Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing ; New York, 1996; pp 212–219. source
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PremisecitationThe unitary e^{i arccos(H/λ)} can be implemented using a quantum walk operator, which requires fewer gates than time evolution.
“The unitary e i arccos( H/λ ) can be implemented using a quantum walk operator 189 which requires fewer gates than time evolution.”
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PremisecitationMCSCF states have non-zero overlap with a polynomial number of computational basis states, for which there are efficient methods of preparation.
“These states have non-zero overlap with a polynomial number of computational basis states, for which there are efficient methods of preparation 288 .”
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InterpretationreasoningBerry et al.'s algorithm is a polynomial and exponential improvement over the previous algorithm.
“These are, respectively, a polynomial and an exponential improvement over the previous algorithm.”
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PremisecitationLow and Wiebe introduced Hamiltonian simulation in the interaction picture.
“Low and Wiebe 197 introduced Hamiltonian simulation in the interaction picture.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [197] source
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Premisenone givenThe success of an algorithm for determining ground state energy depends on the quality of the state preparation.
“The success of an algorithm for determining ground state energy depends on the quality of the state preparation.”
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Premisenone givenThe ground state energy as a function of the nuclear coordinates of a molecular system is sufficient for gleaning many industry-relevant properties.
“In particular, the ground state energy as a function of the nuclear coordinates (the ground state energy manifold) of a molecular system is sufficient for gleaning many of these properties.”
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PremisecitationBerry et al.'s antisymmetrization algorithm achieves a gate count scaling of O(η log η log N) and a circuit depth of O(log η log log N), where η is the number of particles and N ≥ η is the number of single-particle basis functions.
“This algorithm achieves a gate count scaling of O ( η log η log N ) and a circuit depth of O (log η log log N ) where η stands for the number of particles and N ≥ η for the number of single-particle basis functions.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [279] source
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PremisecitationZalka considered discretizing the wave function and initializing simulation through a series of controlled rotations.
“In his pioneering work, Zalka 41 considered discretizing the wave function and initializing simulation through a series of controlled rotations.”
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PremisecitationGrover and Rudolph gave a similar algorithm to prepare a state corresponding to an efficiently integrable probability distribution.
“Grover and Rudolph 285 gave a similar algorithm to prepare a state corresponding to an efficiently integrable probability distribution.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [285] Wang, C.; Wossnig, L. A Quantum Algorithm for Simulating Non-Sparse Hamiltonians. 2018, arXiv:1803.08273 [quant-ph]; https://arxiv.org/abs/1803.08273.
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PremisereasoningA state preparation ansatz must have a significant overlap with the ground state and must be efficiently preparable on a quantum computer.
“Such a state must have two properties. First, the ansatz needs to a have a significant overlap with the ground state. Second, the state has to be efficiently preparable on a quantum computer.”
Quote in the paperSubstantial claimMay combine two claims.
PremisecitationLCU has a query complexity of O(d^2 τ log(d^2 τ_max/ε) log log(d^2 τ_max/ε)), which is optimal in inverse precision.
“LCU has a query complexity of O ( d 2 τ log ( d 2 τ max /ϵ ) log log( d 2 τ max /ϵ ) ), which is optimal in inverse-precision.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [188] source
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Interpretationnone givenThis would be a costly procedure in practice.
“However, this would be a costly procedure in practice.”
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PremisecitationBabbush et al. showed that the electronic structure Hamiltonian requires only O(N^2) terms in a plane wave basis while O(N^4) terms are necessary for Gaussians.
“Babbush et al. 301 showed that the electronic structure Hamiltonian requires only O ( N 2 ) terms in a plane wave basis while O ( N 4 ) terms are necessary for Gaussians.”
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PremisecitationThis allowed the number of qubits in the QPEA ancilla register to be reduced from 20 to 4, enabling the study of quantum algorithms for electronic structure problems on a classical computer.
“This allowed the number of qubits in the QPEA ancilla register to be reduced from 20 to 4, enabling the study of quantum algorithms for electronic structure problems on a classical computer 45 .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [45] source
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PremisecitationBabbush et al.'s algorithm uses quantum signal processing and qubitization to achieve T complexity O(N+log1/ε) for Hamiltonian simulation and O(N^3/ε + N^2 log 1/ε) for energy estimation.
“This algorithm uses quantum signal processing and qubitization to achieve T complexity O ( N +log1 /ϵ ) for Hamiltonian simulation and O ( N 3 /ϵ + N 2 log 1 /ϵ ) for energy estimation.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [281] source
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PremisecitationThis allows a representation of the chemistry Hamiltonian with a number of terms scaling quadratically in the number of spin-orbitals, a significant reduction on the number of terms in the Hamiltonian when using molecular orbitals.
“This allows a representation of the chemistry Hamiltonian with a number of terms scaling quadratically in the number of spin-orbitals, a significant reduction on the number of terms in the Hamiltonian when using molecular orbitals.”
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PremisecitationIndependently of Lloyd's quantum simulation algorithm, Zalka introduced a quantum algorithm for simulation of single- and many-body quantum systems.
“Independently of Lloyd's quantum simulation algorithm 37 , Zalka 273 introduced a quantum algorithm for simulation of single- and many-body quantum systems.”
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PremisecitationBoth Low and Wiebe and Babbush et al. use interaction picture simulations and plane wave basis and achieve O(N^{11/3}) and O(η log N) complexities.
“Both results use interaction picture simulations 197 and plane wave basis and achieve O ( N 11 / 3 ) and O ( η log N ) complexities.”
Quote in the paperNot a substantial claim0 of 2 sources checked- cites [197] source
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- cites [198] Kivlichan, I. D.; Gidney, C.; Berry, D. W.; Wiebe, N.; McClean, J.; Sun, W.; Jiang, Z.; Rubin, N.; Fowler, A.; Aspuru-Guzik, A.; Babbush, R.; Neven, H. Improved Fault Tolerant Quantum Simulation of Condensed-Phase Correlated Electrons via Trotterization. 2019, arXiv:1902.10673 [quant-ph]; https://arxiv.org/abs/1902.10673. source
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PremisecitationWhile the UCC ansatz is intractable to express with a classical computer, a Trotter-Suzuki decomposition approximation of this transformation can be implemented efficiently on a quantum computer.
“While the UCC ansatz is intractable to express with a classical computer 322-324 , a Trotter-Suzuki decomposition approximation (c.f. Section 2.3) of this transformation can be implemented efficiently on a quantum computer 52,320 .”
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- cites [52] Wiesner, S. Simulations of Many-Body Quantum Systems by a Quantum Computer. 1996, arXiv:quant-ph/9603028; https://arxiv.org/abs/quant-ph/9603028.
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- cites [320] source
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PremisecitationThe Hamiltonian can be expressed more easily in first quantization but the basis set necessary for expressing the quantum state is smaller in second quantization.
“They pointed out that the Hamiltonian can be expressed more easily in first quantization but the basis set necessary for expressing the quantum state is smaller in second quantization.”
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Interpretationnone givenIn principle, it is possible to estimate any physical quantity or observable that can be expressed through a low-depth quantum circuit and single-qubit measurements.
“In principle, it is possible to estimate any physical quantity or observable that can be expressed through a low-depth quantum circuit and single-qubit measurements.”
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PremisecitationKivlichan et al. simulated interacting particles and achieved super-polynomially higher accuracy than previous algorithms.
“For real state simulations, Kivlichan et al. 76 simulated interacting particles and achieved super-polynomially higher accuracy than previous algorithms.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [76] Aaronson, S.; Arkhipov, A. The Computational Complexity of Linear Optics. Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing , New York, 2011; pp 333–342. source
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PremisecitationThe pessimistic conclusion of Wecker et al. was that the required spatial resources (number of qubits) were just an order of magnitude more than current devices, but the required time resources demanded a quantum computer whose coherence time was many orders of magnitude beyond current devices.
“The pessimistic conclusion was that, while the required spatial resources (i.e. the number of qubits) were just an order of magnitude more than current devices, the required time resources demanded a quantum computer whose coherence time was many orders of magnitude beyond the capabilities of current devices.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [47] source
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PremisecitationSimulating the complete nuclear and electronic wave function in a chemical reaction of four or more particles is more efficient than using the Born-Oppenheimer approximation.
“They conclude that simulating the complete nuclear and electronic wave function in a chemical reaction of four or more particles is more efficient than using the Born-Oppenheimer approximation.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [75] Aharonov, D.; van Dam, W.; Kempe, J.; Landau, Z.; Lloyd, S.; Regev, O. Adiabatic Quantum Computation Is Equivalent to Standard Quantum Computation. 45th Annual IEEE Symposium on Foundations of Computer Science , 2004.
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PremisecitationFor a second-quantized Hamiltonian, the HF state may be prepared as a product state on the qubits and is therefore easy to prepare.
“For a second-quantized Hamiltonian, the HF state may be prepared as a product state on the qubits and is, therefore, easy to prepare 45 .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [45] source
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PremisecitationStandard quantum chemistry methods can be used to lower-bound the gap even though exactly computing it can be demanding.
“Standard quantum chemistry methods can be used to lower-bound this gap 45 even though exactly computing it can be demanding.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [45] source
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PremisecitationThe algorithm used in Wecker et al.'s analysis was a naive Trotter decomposition, similar to the proposal of Aspuru-Guzik et al., which gives a gate count scaling of O(N^11), and an empirical scaling of O(N^9).
“The algorithm used in their analysis was a 'naive Trotter decomposition', similar to the proposal of Aspuru-Guzik et al. 45 , which gives a gate count scaling of O ( N 11 ), and an empirical scaling of O ( N 9 ).”
Quote in the paperSubstantial claim0 of 2 sources checkedProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
InterpretationreasoningAlthough these proof-of-principle experiments are groundbreaking, it is not clear how to scale them because of their reliance on Hamiltonians simplifications and tomography.
“Although these proof-of-principle experiments are groundbreaking, it is not clear how to scale them because of their reliance on Hamiltonians simplifications and tomography.”
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PremisecitationThe Green's function for a quantum particle is approximated by a product of exponentials with potential and kinetic energy in the exponents.
“The Green's function for a quantum particle is then approximated by a product of exponentials with potential and kinetic energy in the exponents.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [273] Childs, A. M.; Kothari, R. Simulating Sparse Hamiltonians With Star Decompositions, Conference on Quantum Computation, Communication, and Cryptography , 2010; pp 94–103. source
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PremisecitationUCCSD has been demonstrated experimentally for H2 on a superconducting quantum computer.
“Finally, UCCSD has been demonstrated experimentally for HeH + using a single trapped ion 53 , for H 2 on a superconducting quantum computer 55 ,”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [55] source
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PremisecitationUCCSD has been demonstrated experimentally for HeH+ using a single trapped ion.
“Finally, UCCSD has been demonstrated experimentally for HeH + using a single trapped ion 53 , for H 2 on a superconducting quantum computer 55 ,”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [53] source
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Premisenone givenIn the Hamiltonian variational approach, the number S of repetitions of the circuit block can be varied to control the approximation accuracy.
“and S is the number of repetitions of the circuit block and can be varied to control the approximation accuracy.”
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Premisenone givenCCSD(T) suffices to achieve chemical accuracy for energies of molecules near equilibrium.
“A method such as CCSD(T) (where the effect of triple excitations is included using perturbation theory) suffices to achieve chemical accuracy for energies of molecules near equilibrium.”
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PremisecitationAnalytical gradients require orders-of-magnitude fewer measurements than numerical gradients.
“It has been pointed out that analytical gradients require orders-of-magnitude fewer measurements than numerical gradients 329 .”
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InterpretationreasoningThe Hamiltonian variational circuit definition resembles a first-order Trotterized evolution under the Hamiltonian, with terms multiplied by arbitrary parameters.
“The definition of the circuit in the product of Equation 37 resembles a first order Trotterized evolution under the Hamiltonian, where now the terms have been multiplied by arbitrary parameters.”
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PremisecitationThe performance of the Hamiltonian variational approach applied to the Fermi-Hubbard model in a 2D lattice and the electronic structure Hamiltonian is analyzed in Wecker et al.
“The performance of the Hamiltonian variational approach applied to the Fermi-Hubbard model in a 2D lattice and the electronic structure Hamiltonian is analyzed in Wecker et al. 325 .”
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Premisenone givenOther versions of the Hamiltonian variational ansatz can be obtained by mimicking higher-order Trotter-Suzuki formulas.
“Other versions of Equation 37 can be obtained by mimicking higher-order Trotter-Suzuki formulas.”
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InterpretationreasoningIn the case of the Hubbard model, the Hamiltonian variational approach provides an ansatz that scales only linearly with the size of the system and is therefore a good candidate for experimental demonstration in near-term devices.
“In the case of the Hubbard model, it also provides an ansatz that scales only linearly with the size of the system and therefore is a good candidate for experimental demonstration in near-term devices.”
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PremisecitationThe Hamiltonian variational approach, proposed by Wecker et al., defines a strategy for building variational ansatze for arbitrary Hamiltonians.
“The Hamiltonian variational approach, proposed by Wecker et al. 47 , defines a strategy for building variational ansatze for arbitrary Hamiltonians.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [47] source
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Interpretationnone givenThe Hamiltonian variational approach greatly reduces the number of variational parameters, which eases the optimization.
“The Hamiltonian variational approach greatly reduces the number of variational parameters, which eases the optimization.”
Quote in the paperNot a substantial claimMay combine two claims.
PremisereasoningIn a Trotterized implementation of UCCSD, the number of ansatz parameters scales as O(N^2 η^2), which is less than O(N^4).
“In the case of a Trotterized implementation of UCCSD, the ansatz parameters correspond to the cluster amplitudes, whose number scales as O ( N 2 η 2 ) < O ( N 4 ).”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Premisenone givenIf all terms within a subset of the Hamiltonian commute, the corresponding exponential can be implemented exactly.
“If all the terms within a subset were to commute, this could be implemented exactly.”
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InterpretationreasoningUnitary coupled-cluster is unitary and variational, which offers an advantage compared to standard coupled-cluster.
“Consequently the corresponding CC operator is unitary and the formulation is variational, which offers an advantage compared to standard CC.”
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PremisecitationUCCSD has been implemented in combination with different basis set representations of the chemistry Hamiltonian, including the particle/hole basis and the Bogoliubov basis.
“UCCSD has been implemented in combination with different basis set representations of the chemistry Hamiltonian, including the particle/hole basis 327 and the Bogoliubov basis 326 .”
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PremisecitationThe block-by-block optimization procedure improves convergence and is analogous to the segment-by-segment pre-training strategy employed in machine learning to avoid barren plateaus in neural network optimization.
“The block-by-block optimization procedure improves convergence and is analogous to the segment-by-segment pre-training strategy employed in machine learning to avoid barren plateaus in the optimization of neural networks 333,334 .”
Quote in the paperNot a substantial claim0 of 2 sources checkedProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
PremisecitationAdiabatically parametrized states, introduced in McClean et al., parametrize the functions defining the annealing schedule and optimize them using variational minimization.
“A related idea is the concept of adiabatically parametrized states, introduced in McClean et al. 216 , where the functions defining the annealing schedule are parametrized and subsequently optimized using variational minimization.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [216] Frenkel J., 1934
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PremisecitationWecker et al. proposed two optimization strategies for minimizing the energy of the Hamiltonian variational ansatze: global variational and annealed variational.
“Wecker et al. 325 also proposed two different optimization strategies for minimizing the energy of the ansatze defined in Equation 37.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [325] source
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PremisecitationDiscarding operators with low amplitudes based on classical efficient approximations can significantly reduce the number of gates required and provides approximate amplitudes for initializing the optimization, which is critical for improving convergence.
“The last strategy also has the potential to significantly reduce the number of gates required and provides approximate amplitudes for initializing the optimization, which has been pointed out as critical for improving convergence 329 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [329] source
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PremisecitationThe scaling of f with N depends on the fermion-to-qubit mapping: Bravyi-Kitaev gives f in O(log N), while Jordan-Wigner gives f in O(N).
“The scaling of f with N depends on the fermion-to-qubit mapping employed: in the case of the Bravyi-Kitaev transformation f ∈ O (log( N )) while in the case of Jordan-Wigner f ∈ O ( N ) (see Appendix B).”
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InterpretationreasoningDifferent Trotterized versions of UCCSD correspond to different ansatze, and variational optimization tends to compensate the differences between the non-Trotterized and Trotterized versions.
“This can be understood considering that different Trotterized versions of UCCSD correspond to different ansatze and the variational optimization tends to compensate the differences between the non-Trotterized and the Trotterized version, which would have different optimal amplitudes.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Premisenone givenTraditional coupled-cluster methods offer chemically accurate results for a variety of systems, including molecules near equilibrium, and are tractable on classical computers.
“Traditional coupled-cluster (CC) methods have long been considered the 'gold standard' of quantum chemistry because they offer chemically accurate results for a variety of systems, including molecules near equilibrium, and because they are tractable on classical computers.”
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Premisenone givenTo date, there lacks theoretical evidence regarding whether hardware heuristic ansatz approaches can efficiently approximate many-electron wave functions.
“Although these approaches can be implemented more easily than existing PMA on NISQ devices, to date, there lacks theoretical evidence one way or the other regarding their ability to efficiently approximate many-electron wave functions.”
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PremisereasoningThe UCCSD circuit depth can be upper-bounded to O(fN^4) for serial execution, but can be improved depending on hardware connectivity and parallelization.
“Similarly, the circuit depth can be upper-bounded to O ( fN 4 ), which corresponds to a serial execution, however this can be improved depending on the connectivity constraints of the hardware and the ability to parallelize operations in the quantum registers.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.
Premisenone givenThe plane wave dual basis set can be viewed as a discrete variable representation and is particularly suited for periodic systems.
“The plain wave dual basis set can be viewed as a discrete variable representation (DVR) and is particularly suited for periodic systems.”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationnone givenThe Hamiltonian in Equation 44 encompasses the Fermi-Hubbard Hamiltonian and can be readily combined with the Hamiltonian variational strategy to produce a variational ansatz.
“Notice that this Hamiltonian also encompasses the Fermi-Hubbard Hamiltonian and can be readily combined with the Hamiltonian variational strategy to produce a variational ansatz.”
Quote in the paperNot a substantial claimMay combine two claims.
PremisecitationTwo of the new entangling blocks are designed to implement number-conserving operators, more suitable for simulations of molecular systems, and can be implemented natively on superconducting processors with transmon qubits.
“Two of these new blocks are designed to implement number-conserving operators, more suitable for simulations of molecular systems, and can be implemented natively on superconducting processor with transmon qubits 340-342 .”
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PremisecitationKivlichan et al. demonstrated that a single Trotter step of the Hamiltonian in Equation 44 can be implemented with a circuit of depth N using a Fermionic Swap Network, which has a constant improvement with respect to the proposal in Babbush et al. and is conjectured to be optimal.
“Finally, Kivlichan et al. 302 demonstrated that a single Trotter step of the Hamiltonian in Equation 44 can be implemented with a circuit of depth N using a Fermionic Swap Network , which has a constant improvement with respect to the proposal in Babbush et al. 301 and is conjectured to be optimal.”
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Premisenone givenFermionic Gaussian states are completely characterized by a quadratic number of parameters and can therefore be manipulated efficiently on a classical computer.
“The advantage of simulating this family of states is that they are completely characterized by a quadratic number of parameters and therefore, can be manipulated efficiently on a classical computer.”
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PremisecitationThe third type of entangling block introduced by Barkoutsos et al. is similar to the original proposal, with Eent being a circuit with all-to-all CNOT gates and with single-qubit rotations corresponding to Rz.
“The third type of entangling block introduced by Barkoutsos et al. 327 is similar to the original proposal (Equation 68), with E ent being a circuit with all-to-all CNOT gates and with single-qubit rotations corresponding to R z .”
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PremisecitationBarkoutsos et al. proposed a new set of entangling blocks.
“More recently, Barkoutsos et al. 327 proposed a new set of entangling blocks.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [327] source
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Premisenone givenFreeing the T and V terms as variational parameters scales as O(N^2) compared to O(N^4) of the UCCSD ansatz.
“Alternatively, one could free the T and V terms as variational parameters, which scales as O ( N 2 ) compared to O ( N 4 ) of the UCCSD ansatz.”
Quote in the paperNot a substantial claimMay combine two claims.
PremisecitationIn Babbush et al., the idea of using the plane wave dual basis with the Hamiltonian variational strategy is developed to propose a VQE simulation for Jellium that requires depth O(N·S) and optimizes only O(N) parameters by exploiting translational invariance.
“In Babbush et al. 301 , this idea is developed to propose a VQE simulation for Jellium, that requires depth O ( N · S ) and optimizes only over O ( N ) parameters by exploiting the translational invariance of this system.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [301] source
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PremisecitationIn the original paper, the authors applied LDCA to a system with 8 qubits, corresponding to a 2×2 Fermi-Hubbard lattice and an extended Fermi-Hubbard model describing the isomerization of cyclobutadiene.
“In the original paper 326 , the authors applied LDCA to a system with 8 qubits, corresponding to a 2 × 2 Fermi-Hubbard lattice and an extended Fermi-Hubbard model describing the isomerization of cyclobutadiene.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [326] source
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Premisenone givenThe matchgates can be decomposed into a product of two-qubit evolutions under XX, XY, YX, and YY, with the evolution times obtainable from the covariance matrix.
“The matchgates can be decomposed into a product of two-qubit evolutions under XX , XY , Y X , and Y Y , with the evolution times obtainable from the covariance matrix.”
Quote in the paperNot a substantial claimMay combine two claims.
PremisecitationOther linear-depth methods have better constant factors, but their circuits are less general.
“While the other linear-depth methods have better constant factors, their circuits are less general 302,326,335 .”
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- cites [326] source
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- cites [335] Novo, L.; Berry, D. W. Improved Hamiltonian Simulation via a Truncated Taylor Series and Corrections. 2016, arXiv:1611.10033 [quant-ph]; https://arxiv.org/abs/1611.10033.
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Premisenone givenThe LDCA acts on the quasiparticle vacuum state for the Bogoliubov picture, such that the initial states correspond to the generalized Hartree-Fock optimal state for the VQE target Hamiltonian.
“The so-called low depth circuit ansatz (LDCA), acts on the quasiparticle vacuum state for the Bogoliubov picture, such that the initial states correspond to the generalized Hartree-Fock optimal state for the VQE target Hamiltonian.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationHardware efficient ansatze comprise a series of parametrized quantum circuits originally motivated by the limitations of existing NISQ hardware.
“These ansatze comprise a series of parametrized quantum circuits originally motivated by the limitations of existing NISQ hardware 56 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [56] source
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PremisecitationIn the first example of the hardware efficient ansatz, each entangling block comprised a single layer of parametrized single-qubit rotations followed by a fixed set of entangling two-qubit gates specially chosen for a six-qubit quantum processor employed in the experiment.
“In the first example of this ansatz 56 , each entangling block comprised a single layer of parametrized single-qubit rotations followed by a fixed set of entangling two- qubit gates, specially chosen for a six-qubit quantum processor employed in the experiment.”
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PremisecitationDallaire-Demers et al. suggest that a similar circuit template could be used to create a variational ansatz by complementing the list of matchgates with neighboring ZZ interactions, allowing for the preparation of non-Gaussian fermionic states.
“Dallaire-Demers et al. 326 suggest that a similar circuit template could be used to create a variational ansatz by complementing the list of matchgates with neighboring ZZ interactions, allowing for the preparation of non-Gaussian fermionic states.”
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PremisecitationA similar idea has been used to design ansatze for preparing non-Gaussian states efficiently on classical computers, with high success.
“A similar idea has been used to design ansatze for preparing non-Gaussian states efficiently on classical computers, with high success 338 .”
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PremisecitationBoth methods for preparing fermionic Gaussian states achieve the same scaling but differ in the types of gates and the numerical methods employed to compute the angles of the preparation circuits.
“Notice that both methods 326,335 achieve the same scaling but differ in the types of gates and the numerical methods employed to compute the angles of the preparation circuits.”
Quote in the paperNot a substantial claim0 of 2 sources checked- cites [326] source
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- cites [335] Novo, L.; Berry, D. W. Improved Hamiltonian Simulation via a Truncated Taylor Series and Corrections. 2016, arXiv:1611.10033 [quant-ph]; https://arxiv.org/abs/1611.10033.
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Interpretationnone givenThe LDCA has the advantage of being able to treat systems where the number of particles is not conserved, such as superconductors, and therefore expands the range of applications of VQE to problems in condensed matter physics.
“The LDCA also has the advantage of being able to treat systems where the number of particles is not conserved, such as superconductors, and therefore expands the range of applications of VQE to problems in condensed matter physics.”
Quote in the paperSubstantial claimMay combine two claims.
PremisecitationFrom the covariance matrix it is possible to obtain the corresponding W matrix, which is later decomposed into 2N(N-1) matchgates acting on neighboring qubits and N local rotations, using the Hoffmann algorithm.
“From the covariance matrix it is possible to obtain the corresponding W matrix, which is later decomposed into 2 N ( N -1) matchgates acting on neighboring qubits and N local rotations, using the Hoffmann algorithm 337 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [337] source
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Premisenone givenThe LDCA also implements XY and YX interactions, which do not conserve the number of particles.
“This is very similar to the construction of the LDCA circuit with the difference that LDCA also implements XY and Y X interactions, which do not conserve the number of particles.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Interpretationnone givenThe fermionic swap network circuit could be readily employed in combination with the Hamiltonian variational strategy to create a variational ansatz for preparing ground states of the Hamiltonian of Equation 44 using VQE.
“The fermionic swap network circuit could be readily employed in combination with the Hamiltonian variational strategy to create a variational ansatz for preparing ground states of the Hamiltonian of Equation 44 using VQE.”
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Premisenone givenThe circuits for the fermionic swap network comprise fermionic swap gates applied on neighboring qubits that implement evolution under ZZ, Z, XX and YY.
“The circuits for the fermionic swap network comprise fermionic swap gates applied on neighboring qubits that implement evolution under ZZ , Z , XX and Y Y .”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationThe low depth circuit ansatz proposed by Dallaire-Demers et al. is inspired by a circuit for preparing fermionic Gaussian states on quantum computers.
“Another ansatz for state preparation in VQE is the so-called low depth circuit ansatz (LDCA) proposed by Dallaire-Demers et al. 326 . This ansatz is inspired by a circuit for preparing fermionic Gaussian states (FGSs) on quantum computers.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [326] source
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InterpretationreasoningThere are similarities between the LDCA circuit and the fermionic swap network circuit proposed in the context of Hamiltonian evolution for Equation 44.
“In the same vein we note some similarities between the LDCA circuit and the fermionic swap network circuit proposed in the context of Hamiltonian evolution for Equation 44.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationIn this first hardware efficient ansatz example, the total number of parameters scales as 3N(L+2).
“In this case, the total number of parameters scales as 3 N ( L + 2).”
Quote in the paperNot a substantial claimMay combine two claims.
PremisecitationChoosing ε_i^2 proportional to |h_i| reduces the estimated number of measurements, and this choice can be proved optimal by application of the Lagrange conditions.
“A more practical approach is to choose ϵ 2 i to be proportional to | h i | . This reduces the estimate in the number of measurements to:”
- Also stated in: § 5.1.2 Energy estimation in VQE · p. 102 · plain
“which can be proved to be optimal by application of the Lagrange conditions 346 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [346] source
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- Also stated in: § 5.1.2 Energy estimation in VQE · p. 102 · plain
Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
PremisecitationSeveral numerical studies comparing the efficacy of different optimization methods have been published since VQE was first introduced; the original paper used Nelder-Mead, McClean et al. found LGO usually superior, and Romero et al. found COBYLA and L-BFGS-B showed better performance than Powell and Nelder-Mead for H4.
“Several numerical studies comparing the efficacy of different optimization methods have been published since VQE was first introduced; the original paper 52 used Nelder-Mead, a simple derivative-free optimization algorithm.”
- Also stated in: § 5.1.3 Optimization methods · p. 107 · plain
“McClean et al. 216 studied VQE for optimizing H 2 using the software package TOMLAB 351 to compare four algorithms: Nelder-Mead, MULTMIN, GLCCLUSTER, and LGO, finding that LGO was usually superior.”
- Also stated in: § 5.1.3 Optimization methods · p. 107 · plain
“Romero et al. 329 simulated VQE for H 4 (a system of four hydrogen atoms), using both gradient and derivativefree methods for the classical optimization step. They used algorithms known as L-BFGS-B, COBYLA, Powell, and Nelder-Mead, observing that the derivative-free methods Powell and Nelder-Mead often had difficulties converging to the correct answer in the large parameter space required by VQE,”
Quote in the paperNot a substantial claim0 of 3 sources checked- cites [52] Wiesner, S. Simulations of Many-Body Quantum Systems by a Quantum Computer. 1996, arXiv:quant-ph/9603028; https://arxiv.org/abs/quant-ph/9603028.
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- cites [216] Frenkel J., 1934
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- cites [329] source
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- Also stated in: § 5.1.3 Optimization methods · p. 107 · plain
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Premisenone givenAny Hermitian operator, such as the electronic structure Hamiltonian, can be decomposed in the basis of Pauli operators.
“The basic idea behind this approach is that any Hermitian operator, such as the electronic structure Hamiltonian, can be decomposed in the basis of Pauli operators.”
Quote in the paperNot a substantial claimMay combine two claims.
PremisecitationIn VQE applied to electronic structure Hamiltonians, analytical gradients offer a practical advantage over numerical gradients, and numerical gradients require a few orders of magnitude more measurements to achieve the same accuracy.
“This is in sharp contrast with VQE applied to electronic structure Hamiltonians 329 , where the conclusion is that analytical gradients offer a practical advantage over numerical gradients. The numerical gradients require a few orders of magnitude more measurements to achieve the same accuracy.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [329] source
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PremisecitationAnalysis applied to Hamiltonian averaging showed that the number of measurements required for achieving precision ε in the energy is expected to scale as O(N^6/ε^2) for molecules of sufficient size, compared to the formal O(N^8/ε^2) estimated based on the number of terms in the Hamiltonian.
“This analysis applied to Hamiltonian averaging showed that the number of measurements required for achieving precision ϵ in the energy is expected to scale as O ( N 6 /ϵ 2 ) for molecules of sufficient size, compared to the formal O ( N 8 /ϵ 2 ) estimated based on the number of terms in the Hamiltonian 306 .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [306] Tubman, N. M.; Mejuto-Zaera, C.; Epstein, J. M.; Hait, D.; Levine, D. S.; Huggins, W.; Jiang, Z.; McClean, J. R.; Babbush, R.; Head-Gordon, M.; Whaley, K. B. Postponing the Orthogonality Catastrophe: Efficient State Preparation for Electronic Structure Simulations on Quantum Devices. 2018, arXiv:1809.05523 [quant-ph]; https://arxiv.org/abs/1809.05523.
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Premisenone givenOnce successfully trained, the encoding circuit of a quantum autoencoder can compress a set of n-qubit states to m-qubit states, where m ≤ n.
“Once successfully trained, the encoding circuit of the overall QAE circuit can compress a set of n -qubit states to m -qubit states, where m ≤ n .”
Quote in the paperSubstantial claimClassified as probably not a substantial claim.
PremisecitationChemical accuracy is a standard for the desired energy accuracy of 1 kcal/mol (1.59 × 10^-3 Hartrees, 43.3 meV) or less.
“This sensitivity motivates the concept of chemical accuracy, which is a standard for the desired energy accuracy of 1 kcal/mol (1 . 59 × 10 -3 Hartrees, 43 . 3 meV) or less 79 .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [79] source
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- Rests on an assumption: Reaction rates are exponentially sensitive to changes in energy; a change of only 1.4 kcal/mol in the free energy generates a change of one order of magnitude in the estimate of a reaction rate.
Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Interpretationnone givenThe choice or design of the ansatz is a crucial component for algorithmic performance.
“As we have reviewed earlier in this section, the choice/design of the ansatz is a crucial component for algorithmic performance.”
Quote in the paperNot a substantial claimMay combine two claims.
PremisecitationMost experimental implementations so far have employed gradient-free methods that better tolerate fluctuations, such as Nelder-Mead and Particle-Swarm Optimization, and more recently SPSA has been applied.
“Correspondingly, most of the experimental implementations so far have employed methods that better tolerate these fluctuations, such as Nelder-Mead and Particle-Swarm Optimization (PSO) 353 , which are gradient free. More recently the Simultaneous Perturbation Stochastic Approximation (SPSA) has been applied 354,355 .”
Quote in the paperNot a substantial claim0 of 3 sources checkedMay combine two claims.
InterpretationcitationThe recent observation that the variational parameter landscape is often plagued by barren plateaus implies that care must be taken when designing a circuit ansatz for VQE; by Levy's lemma, expectation values only substantially deviate from their mean over an exponentially small fraction of the Hilbert space, so arbitrary or unstructured ansatze may be uninformative and expensive.
“The recent observation 321 that the variational parameter landscape is often plagued by the overabundance of 'barren plateaus' implies that care must be taken when designing a circuit ansatz for the VQE algorithm. By Levy's lemma, one can prove that expectation values only substantially deviate from their mean over an exponentially small fraction of the Hilbert space. This implies that the use of”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [321] source
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Premisenone givenThe kinetic operator is not diagonal in the plane wave dual basis, but it can be made diagonal by the application of a fermionic fast Fourier transform (FFFT).
“The kinetic operator is not diagonal in the plane wave dual basis, but it can be made diagonal by the application of a fermionic fast Fourier transform (FFFT).”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationPeruzzo et al. suggested Hamiltonian averaging as an alternative scheme for computing the expectation value of local Hamiltonians.
“To avoid the overhead of PEA, Peruzzo et al. 52 suggested an alternative scheme called Hamiltonian averaging for computing the expectation value of local Hamiltonians.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [52] Wiesner, S. Simulations of Many-Body Quantum Systems by a Quantum Computer. 1996, arXiv:quant-ph/9603028; https://arxiv.org/abs/quant-ph/9603028.
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Interpretationnone givenThe computational cost of operator averaging is one of the factors contributing to the overall cost of VQE prediction, the others being the depth of the circuit for state preparation and the number of function calls required by the optimization routine.
“The computational cost of operator averaging is one of the factors contributing to the overall cost of VQE prediction, the others being the depth of the circuit for state preparation and the number of function calls required by the optimization routine.”
Quote in the paperNot a substantial claimMay combine two claims.
PremisecitationGrouping commuting terms in the Hamiltonian reduces the number of state preparations required for a single estimate of the energy, and commuting terms can be measured in sequence without biasing the final expectation.
“A second strategy is to group commuting terms in the Hamiltonian to reduce the number of state preparations required for a single estimate of the energy, which was first described in McClean et al. 216 .”
- Also stated in: § Strategies for reducing the number of measurements in Hamiltonian averaging. · p. 105 · plain
“Commuting terms can be measured in sequence without biasing the final expectation.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [216] Frenkel J., 1934
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- Also stated in: § Strategies for reducing the number of measurements in Hamiltonian averaging. · p. 105 · plain
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Premisenone givenThe wave function of molecular hydrogen in the minimal basis at a particular bond length requires four qubits to represent the occupancy of the ground state.
“For instance, the wave function of molecular hydrogen (in the minimal basis) at a particular bond length requires four qubits to represent the occupancy of the ground state.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationTerm truncation is a strategy to reduce the cost of operator averaging by ordering Hamiltonian terms according to the norm of their coefficients and removing the k* lowest terms such that δ_k* < Cε^2.
“The first of these approaches is term truncation, briefly described in the previous section. Implementation of this approach 216,306 requires ordering the Hamiltonian terms according to the norm of their coefficients, which correspond to the maximum expected contribution to the total expectation value.”
Quote in the paperNot a substantial claim0 of 2 sources checked- cites [216] Frenkel J., 1934
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- cites [306] Tubman, N. M.; Mejuto-Zaera, C.; Epstein, J. M.; Hait, D.; Levine, D. S.; Huggins, W.; Jiang, Z.; McClean, J. R.; Babbush, R.; Head-Gordon, M.; Whaley, K. B. Postponing the Orthogonality Catastrophe: Efficient State Preparation for Electronic Structure Simulations on Quantum Devices. 2018, arXiv:1809.05523 [quant-ph]; https://arxiv.org/abs/1809.05523.
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Interpretationnone givenIn experimental implementations of VQE on NISQ devices, the accuracy of the optimization can be significantly lowered by fluctuations on the measured properties caused by noise and finite measurement statistics, and this is a major hurdle to the implementation of VQE for larger systems.
“In experimental implementations of VQE on NISQ devices, the accuracy of the optimization can be significantly lowered by the fluctuations on the measured properties caused by noise and finite measurement statistics. This is a major hurdle to the implementation of VQE for larger systems, where the number of parameters can be significant even for linear scaling ansatze.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Premisenone givenVQE relies on ultimately preparing and then measuring an entangled state of many qubits.
“For example, VQE relies on ultimately preparing (and then measuring) an entangled state of many qubits.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationN-representability conditions on the fermionic reduced density matrices can be exploited to reduce the number of measurements in VQE calculations.
“More recently, Rubin et al. 346 pointed out that N-representability conditions on the fermionic reduced density matrices can be exploited to reduce the number of measurements in VQE calculations.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [346] source
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InterpretationcitationThe shorter ansatz generated by CUSP is less vulnerable to noise on a NISQ computer, resulting in more accurate VQE energies.
“CUSP uses an existing VQE circuit ansatz, together with a quantum autoencoder, to generate a new ansatz that is shorter in gate depth and hence less vulernable to noise on a NISQ computer, resulting in more accurate VQE energies.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [345] source
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Premisenone givenNot all Hamiltonian terms can be measured simultaneously, which increases the number of measurements needed to achieve a given accuracy.
“However, not all the Hamiltonian terms can be measured simultaneously, which increases the number of measurements needed to achieve a given accuracy.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
InterpretationreasoningThe chemical accuracy requirement sets an upper bound to the precision ε for energy estimation in VQE and in general to any quantum simulation algorithm for chemistry.
“This requirement sets an upper bound to the precision, ϵ , for energy estimation in VQE and in general to any quantum simulation algorithm for chemistry.”
Quote in the paperNot a substantial claimMay combine two claims.
PremisereasoningThe effective number of measurements depends on the magnitude of the Hamiltonian coefficients, which for electronic structure correspond to linear combinations of 1-electron and 2-electron integrals, which in turn depend on the choice of basis set.
“Therefore, the effective number of measurements will be dependent on the magnitude of the Hamiltonian coefficients, which is the case of electronic structure correspond to linear combinations of 1electron and 2-electron integrals, which in turn depend on the choice of basis set for the calculation”
Quote in the paperSubstantial claimMay combine two claims.
PremisecitationFor Hamiltonians in the plane wave dual basis, the number of measurements required to estimate the expectation values can be bounded, and further savings are possible because the interaction (V) and potential (U) terms are diagonal and commute, allowing separate unbiased estimation for the mean of their sum.
“Alternatively, for Hamiltonians in the plane wave dual basis (Equation 44), the number of measurements required to estimate the expectation values can be bounded as 301”
- Also stated in: § 5.1.2 Energy estimation in VQE · p. 103 · hedged
“In this case, it is possible to obtain further savings in the number of measurements by exploiting the fact of the interaction ( V ) and potential terms ( U ) of the Hamiltonian are diagonal and commute with each other. Therefore they allow the use of a separate, unbiased estimation for the mean of the sum of these terms 316 .”
Quote in the paperSubstantial claim0 of 2 sources checked- Also stated in: § 5.1.2 Energy estimation in VQE · p. 103 · hedged
May combine two claims. The quote only partly states this claim.
InterpretationreasoningThe Adiabatically Assisted Variational Quantum Eigensolver (AAVQE) is inspired by adiabatic evolution, uses the interpolation parameter to adjust the Hamiltonian from one VQE run to the next, and appears to be a promising approach for circumventing the barren plateau issue.
“Finally, an alternative strategy to break the optimization into easier, smaller parts is the Adiabatically Assisted Variational Quantum Eigensolver (AAVQE) method 359 . This method is inspired by the adiabatic evolution of a ground state in quantum annealing.”
- Also stated in: § 5.1.3 Optimization methods · p. 111 · hedged
“Thus, AAVQE, which runs VQE while incrementally adjusting the optimization landscape, appears to be another promising approach for circumventing the barren plateau issue.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [359] source
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- The authors state this as an assumption: In AAVQE, the gap of H(s) does not become too small.
- Also stated in: § 5.1.3 Optimization methods · p. 111 · hedged
May combine two claims.
PremisecitationElectronic structure Hamiltonians by definition have O(N^4) terms; in a local basis the number of non-negligible terms scales as O(N^2), and advanced techniques can evaluate these contributions in O(N) time.
“As noted before, the electronic structure Hamiltonians by definition have O ( N 4 ) terms, however, in a local basis the number of non-negligible terms scale as O ( N 2 ) and advanced techniques can evaluate these contributions in O ( N ) time. This is the basis of linear scaling methods employed in quantum chemistry 79,347 .”
Quote in the paperSubstantial claim0 of 2 sources checkedProbably not a substantial claim: may not pass S3 (load-bearing).
Premisenone givenThe PEA approach provides an estimate of the energy in a single state preparation with cost O(1/ε) for error tolerance ε.
“One possible approach to estimating the energy is the PEA approach, which provides an estimate of the energy in a single state preparation with cost O ( 1 ϵ ) for error tolerance ϵ .”
Quote in the paperNot a substantial claimMay combine two claims.
PremisecitationThe first experimental demonstration of VQE was performed in 2014; the first scalable demonstration of both IPEA and VQE employed three superconducting qubits for simulating H2 in a minimal basis, and VQE with a hardware-based ansatz for H2, LiH and BeH2 was demonstrated in a six-qubit system.
“The first experimental demonstration of VQE was performed in 2014 52 .”
- Also stated in: § 5.1.4 Physical realizations of VQE · p. 112 · plain
“The first scalable demonstration of both the IPEA and VQE algorithms employed three superconducting qubits for simulating H 2 in a minimal basis and was carried out by a Google Research and Harvard collaboration 55 .”
- Also stated in: § 5.1.4 Physical realizations of VQE · p. 112 · boosted
“These demonstrations were followed by the demonstration of VQE with a hardware-based ansatz for H 2 , lithium hydride (LiH) and beryllium hydride (BeH 2 ) in a system with six qubits 56 by the IBM quantum computing team.”
Quote in the paperNot a substantial claim0 of 3 sources checked- Also stated in: § 5.1.4 Physical realizations of VQE · p. 112 · plain
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PremisecitationThe procedure proposed by Rubin et al. adds N-representability constraints to the original Hamiltonian to reduce the sum in Equation 73, equivalent to running VQE for an alternative Hamiltonian, and its application to example molecular Hamiltonians showed reductions of around one order of magnitude in the number of measurements required.
“The basic idea of the procedure proposed in Rubin et al. 346 is to add the constraints to the original Hamiltonian to reduce the sum in Equation 73.”
- Also stated in: § Strategies for reducing the number of measurements in Hamiltonian averaging. · p. 107 · plain
“The application of this strategy to example molecular Hamiltonians showed reductions of around one order of magnitude in the number of measurements required 346 .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [346] source
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- Also stated in: § Strategies for reducing the number of measurements in Hamiltonian averaging. · p. 107 · plain
Probably not a substantial claim: may not pass S4 (based). May combine two claims.
Premisenone givenTo guarantee the success of PEA estimation, all operations must be executed fully coherently on the quantum computer, which is only possible using quantum error correction.
“Unfortunately, to guarantee the success of PEA estimation, all the operations must be executed fully coherently on the quantum computer, which is only possible using quantum error correction.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationCanonical transformation theory and perturbative explicit correlated corrections are other techniques that could be implemented using the strategy of combining VQE with classical algorithms.
“Other techniques that could be implemented using this strategy are canonical transformation theory 119 and perturbative explicit correlated corrections 363,364 .”
Quote in the paperNot a substantial claim0 of 3 sources checked- cites [119] source
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- cites [363] Bravyi, S.; Gambetta, J. M.; Mezzacapo, A.; Temme, K. Tapering Off Qubits to Simulate Fermionic Hamiltonians. 2017, arXiv:1701.08213 [quant-ph]; https://arxiv.org/abs/1701.08213.
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- cites [364] source
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Premisenone givenIn the variational quantum simulator, matrix inversion and multiplication is carried out by a classical processor.
“This matrix inversion and multiplication is carried out by a classical processor.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [200] source
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InterpretationreasoningAn adiabatic quantum computer would evolve the system from an easy Hamiltonian with known ground state to the spin Hamiltonian of interest, thereby encoding the ground state of the molecule in its final state.
“It would then evolve the system from an 'easy' Hamiltonian (such as the transverse field H 0 = ∑ i X i ) with known ground state to the spin Hamiltonian of interest, therefore encoding the ground state of the molecule in its final state.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
InterpretationreasoningThere are no guarantees on the performance of these heuristic algorithms.
“There are no guarantees on their performance, and their proper functioning can only be established on a case-by-case basis.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationA cumulant approach that approximates the energy correction using only the 1-RDM and 2-RDM could be employed to avoid the measurement cost of estimating the 3-RDM and 4-RDM.
“To avoid the measurement cost of estimating these RDMs, a cumulant approach that approximates this correction using only the 1-RDM and 2-RDM could be employed 362 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [362] source
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PremisecitationVQE calculations on active spaces can be combined with multi-reference perturbation theory techniques implemented on a classical computer to add dynamical correlation to the calculations.
“One possibility is to combine VQE calculations on active spaces with multi-reference perturbation theory techniques implemented on a classical computer to add dynamical correlation to the calculations, as suggested in 346 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [346] source
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PremisecitationAn alternative analog approach to quantum computing is based on adiabatic quantum evolution.
“However, there is an alternative 'analog' approach to quantum computing based on adiabatic quantum evolution 60 that has also gained much attention in the quantum information community.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [60] Preskill, J. Quantum Computing in the NISQ Era and Beyond. 2018, arXiv:1801.00862 [quant-ph]; https://arxiv.org/abs/1801.00862. source
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PremisereasoningSome molecular properties, such as the dipole moment and the charge density, can be computed by measuring the corresponding operators represented in the basis set employed for VQE, and the expectation values of 1-electron and 2-electron operators can be calculated using the same 1-RDM and 2-RDM measured for the optimal state obtained from VQE.
“Some of these properties, such as the dipole moment and the charge density, can be computed by measuring the corresponding operators represented in the basis set employed for the VQE.”
- Also stated in: § 5.1.6 Calculation of molecular properties with VQE · p. 116 · plain
“For example, the expectation values of 1-electron ( E 1 ) and 2-electron ( E 2 ) operators can be calculated using the same 1-RDM and 2-RDM measured for the optimal state obtained from VQE:”
Quote in the paperNot a substantial claim- Also stated in: § 5.1.6 Calculation of molecular properties with VQE · p. 116 · plain
Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Interpretationnone givenThe alternative algorithms (variational quantum simulator, imaginary-time VQS) are heuristic, as is VQE.
“An important caveat regarding these alternative algorithms is that, as with VQE, they are heuristic.”
Quote in the paperNot a substantial claimMay combine two claims.
PremisecitationThe Quantum Subspace Expansion (QSE) extends VQE and requires additional measurements to estimate excited state energies; after obtaining the ground state with VQE, an approximate subspace of low-energy excited states is found by linear combinations O_i|ψ〉, and the classical computation diagonalizes the matrix to find the excited state energies.
“In summary, this framework, called the Quantum Subspace Expansion (QSE), extends the VQE algorithm and requires additional measurements to estimate the excited state energies.”
- Also stated in: § 5.1.5 VQE for excited states · p. 114 · plain
“That is, after obtaining the the ground state | ψ 〉 of a molecule using VQE, an approximate subspace of low-energy excited states is found by taking the linear combinations of states of the form O i | ψ 〉 , where the O i are physically motivated quantum operators.”
- Also stated in: § 5.1.5 VQE for excited states · p. 114 · plain
“The classical computation then diagonalizes the matrix to find the excited state energies.”
Quote in the paperNot a substantial claim0 of 2 sources checked- cites [320] source
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- cites [58] Poulin D., 2014, Quantum Inf. Comput.
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- Also stated in: § 5.1.5 VQE for excited states · p. 114 · plain
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PremisecitationThe folded spectrum method utilizes a variational method to converge to the eigenvector closest to a shift parameter λ by minimizing (H - λI)^2, and this methodology requires a quadratic increase in the number of terms of the effective Hamiltonian and a significant increase in the number of measurements needed.
“The first and perhaps the simplest extension consists of the application of the folded spectrum method, which utilizes a variational method to converge to the eigenvector closest to a shift parameter λ .”
- Also stated in: § 5.1.5 VQE for excited states · p. 113 · plain
“This is achieved by variationally minimizing the operator H λ = ( H -λI ) 2 according to Peruzzo et al. 52 . This methodology, though relatively straightforward to implement, requires a quadratic increase in the number of terms of the effective Hamiltonian.”
Quote in the paperNot a substantial claim0 of 2 sources checked- cites [52] Wiesner, S. Simulations of Many-Body Quantum Systems by a Quantum Computer. 1996, arXiv:quant-ph/9603028; https://arxiv.org/abs/quant-ph/9603028.
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- cites [316] source
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- Also stated in: § 5.1.5 VQE for excited states · p. 113 · plain
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Premisenone givenThe results discussed so far are based on gate model quantum computers, which can be broadly categorized as the digital approach to quantum simulation.
“The results discussed so far are based on gate model quantum computers, which can be broadly categorized as the 'digital' approach to quantum simulation.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S4 (based).
Premisenone givenThermal fluctuations which drive the system out of the ground state can be mitigated by keeping the system at extremely low temperatures.
“Thermal fluctuations which drive the system out of the ground state can be mitigated by keeping the system at extremely low temperatures.”
Quote in the paperNot a substantial claimMay combine two claims.
PremisecitationIn the variational quantum simulator, the role of the quantum computer is to explore families of output states and to use these states to efficiently estimate the entries of M and V.
“The role of the quantum computer is to explore families of output states, U ( ⃗ θ ) | ψ 0 〉 , which are unlikely to be representable on a classical computer, and to use these states to efficiently estimate the entries of M and ⃗ V .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [200] source
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Interpretationnone givenA suitable adiabatic quantum computer would be able to effectively encode such a spin Hamiltonian.
“A suitable adiabatic quantum computer would be able to effectively encode such a spin Hamiltonian.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationThe variational quantum simulator algorithm applies a series of tunable gates to prepare a tunable output state.
“Instead of applying quantum gates which simulate Hamiltonian dynamics, the variational quantum simulator algorithm applies a series of tunable gates to prepare a tunable output state.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [200] source
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PremisecitationWAVES utilizes VQE as a subroutine to compute the ground state; its objective function includes the energy and an approximation for the entropy, a control ancilla qubit acts as an eigenstate witness, and a tunable parameter T is used to bias towards excited states.
“An alternative protocol that also utilizes VQE as a subroutine to compute the ground state is the witness-assisted variational eigenspectra solver (WAVES) 316 .”
- Also stated in: § 5.1.5 VQE for excited states · p. 115 · plain
“The objective function in WAVES is augmented to include the energy ( E ) as well as an approximation for the entropy (a purity term Tr[ ρ 2 C ]):”
- Also stated in: § 5.1.5 VQE for excited states · p. 115 · plain
“A tunable parameter T (that can be pre-optimized) is used to bias towards excited states.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [316] source
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- Also stated in: § 5.1.5 VQE for excited states · p. 115 · plain
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PremisecitationProposals exist for simulating chemical reactions and protein folding using adiabatic quantum computing.
“In the context of quantum chemistry, research has focused on the analog simulation of molecules by mapping the electronic structure problem to a Hamiltonian of spins with additional coupling terms 366 , although proposals exist for simulating chemical reactions 367-369 and protein folding simulations 370,371 .”
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PremisereasoningVQE can be used as a subroutine for energy calculation and interfaced with classical routines for geometry optimization and calculation of thermodynamic properties.
“Similarly, the VQE algorithm can be used as a subroutine for energy calculation and interfaced with classical routines for geometry optimization and calculation of thermodynamic properties.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
InterpretationreasoningThe proper functioning of these heuristic algorithms can only be established on a case-by-case basis.
“There are no guarantees on their performance, and their proper functioning can only be established on a case-by-case basis.”
Quote in the paperNot a substantial claimMay combine two claims.
InterpretationreasoningTwo major challenges in successfully implementing adiabatic evolution are keeping the energy scales well below the size of energy gaps in the Hamiltonian throughout the evolution and realizing the many-body interactions.
“Two major challenges in successfully implementing this adiabatic evolution are keeping the energy scales well below the size of energy gaps in the Hamiltonian throughout the evolution and realizing the many-body interactions.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
InterpretationreasoningThe use of quantum error correction stands to improve the performance of variational quantum algorithms such as VQE.
“The use of quantum error correction also stands to improve the performance of variational quantum algorithms such as VQE.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
InterpretationreasoningError correction could lead to an improvement in the preparation of good ansatz states for the variational quantum eigensolver.
“For instance, error-correction could lead to an improvement in the preparation of good ansatz states for the variational quantum eigensolver, as described in Section 5.1.1.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationThe quantum subspace expansion technique, developed in McClean et al. 320, was motivated by the concept of linear response of the ground state.
“The quantum subspace expansion technique, developed in McClean et al. 320 , was motivated by the concept of linear response of the ground state.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [320] source
Not checked in the free run.
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Premisenone givenThe goal of the linear-optics algorithm is to compute the FCP by sampling from the output distribution of a modified boson sampling device.
“The goal is to compute the FCP by sampling from the output distribution of a modified boson sampling device, as shown in Figure 9.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
InterpretationreasoningLinear optics is the most natural platform for the simulation of vibrational quantum dynamics, as photons have the same boson statistics as the vibrational excitations (phonons).
“Linear optics is the most natural platform for the simulation of vibrational quantum dynamics, as photons have the same boson statistics as the vibrational excitations (phonons).”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationIn the past few years, several papers have proposed new quantum algorithms along with experimental demonstrations on quantum devices.
“In the past few years, several papers have proposed new quantum algorithms along with experimental demonstrations on quantum devices 387,388 .”
Quote in the paperNot a substantial claim0 of 2 sources checkedMay combine two claims.
Premisenone givenBosonic dynamics govern many aspects of chemical systems; for instance, the vibrational modes of molecules can be described by phonon transitions, which obey bosonic statistics.
“However, bosonic dynamics govern many other aspects of chemical systems; for instance, the vibrational modes of molecules can be described by phonon transitions, which obey bosonic statistics.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationA recent paper initiated the use of low-rank tensor decompositions for improving the performance of certain quantum algorithms for chemistry simulation.
“For example, a recent paper 381 initiated the use of low-rank tensor decompositions for improving the performance of certain quantum algorithms for chemistry simulation.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [381] Higgott, O.; Wang, D.; Brierley, S. Variational Quantum Computation of Excited States. 2018, arXiv:1805.08138 [quant-ph]; https://arxiv.org/abs/1805.08138. source
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PremisecitationSeveral methods have been introduced which aim to bridge the gap between error-corrected and non-error-corrected quantum algorithms.
“Already, several methods have been introduced which aim to bridge the gap between these two classes of quantum algorithms 382-384 .”
Quote in the paperNot a substantial claim0 of 3 sources checkedProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationThe proposal by Olivares et al. 379 consists of superconducting resonators interacting through tunable couplers.
“In particular the proposal by Olivares et al. 379 consists of superconducting resonators interacting through tunable couplers.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [379] McArdle, S.; Endo, S.; Aspuru-Guzik, A.; Benjamin, S.; Yuan, X. Quantum Computational Chemistry. 2018, arXiv:1808.10402 [quant-ph]; https://arxiv.org/abs/1808.10402.
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PremisecitationWith the arrival of usable quantum devices, we have now entered a prototype era in the field of quantum algorithms.
“With the arrival of usable quantum devices 385,386 , we have now entered a prototype era in the field of quantum algorithms.”
Quote in the paperNot a substantial claim0 of 2 sources checkedProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Premisenone givenSuperconducting circuits have no experimental realizations for simulating molecular dynamics as of yet, but promising architectures have been theoretically proposed.
“Superconducting circuits have no experimental realizations as of yet, however promising architectures have been theoretically proposed.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationIn Olivares et al.'s approach, a quantum quench in the superconducting simulator initially prepared in the molecular ground state causes the system to relax to the ground state populating the resonators with a distribution that resembles the FCP.
“In their approach the authors perform a quantum quench in the superconducting simulator, initially prepared in the molecular ground state. The abrupt change takes the system out of equilibrium, in a way that the system relaxes to the ground state populating the resonators with a distribution that resembles that of the FCP.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [379] McArdle, S.; Endo, S.; Aspuru-Guzik, A.; Benjamin, S.; Yuan, X. Quantum Computational Chemistry. 2018, arXiv:1808.10402 [quant-ph]; https://arxiv.org/abs/1808.10402.
Not checked in the free run.
Classified as probably not a substantial claim.
Premisenone givenHamiltonian simulation serves as a subroutine for other algorithms, such as the quantum phase estimation algorithm.
“But, more importantly, Hamiltonian simulation serves as a subroutine for other algorithms, such as the quantum phase estimation algorithm.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationIn Sparrow et al. 376, the authors simulate the vibrational quantum dynamics of a variety of molecules (H2CS, SO3, HNCO, HFHF, N4 and P4), both in the harmonic and anharmonic regime, using a reprogrammable photonic chip.
“In Sparrow et al. 376 , the authors simulate the vibrational quantum dynamics of a variety of molecules (H 2 CS, SO 3 , HNCO, HFHF, N 4 and P 4 ), both in the harmonic and anharmonic regime, using a reprogrammable photonic chip.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [376] source
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationIn the few years since its invention, there has been significant algorithmic development surrounding the variational quantum eigensolver algorithm.
“In the few years since its invention, there has been significant algorithmic development surrounding the variational quantum eigensolver algorithm 320,326,327,360 .”
Quote in the paperNot a substantial claim0 of 4 sources checkedMay combine two claims.
Interpretationnone givenThe quantum phase estimation method requires the use of quantum error correction to properly function, and is therefore not feasible using near-term quantum devices.
“Although promising, this method requires the use of quantum error correction to properly function, and is therefore not feasible using near-term quantum devices.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Premisenone givenThe second method for estimating ground state energies, the variational quantum eigensolver, was specifically developed to be deployable on currently available quantum hardware without a dependence on quantum error correction for proper functioning.
“In contrast, the second method for estimating ground state energies was specifically developed to be deployable on currently available quantum hardware, without a dependence on quantum error correction for proper functioning.”
Quote in the paperNot a substantial claimMay combine two claims.
InterpretationreasoningOptical quantum systems are native to bosonic statistics and represent a platform for carrying out simulation of bosonic chemical systems more efficiently than encoding the problem into a qubit formalism.
“While one could imagine devising a gate-model quantum algorithm to simulate this system, optical quantum systems are native to bosonic statistics and represent a platform for carrying out this simulation more efficiently than encoding the problem into a qubit formalism.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationOther physical platforms such as trapped ions or superconducting circuits have proven useful in simulating molecular dynamics.
“Nevertheless, other physical platforms such as trapped ions or superconducting circuits have proven useful in simulating molecular dynamics.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S4 (based).
Premisenone givenMolecular Hamiltonians and the adiabatic evolution require the spread of coupling term strengths over multiple orders of magnitude.
“This poses a challenge as molecular Hamiltonians and the adiabatic evolution require the spread of coupling term strengths over multiple orders of magnitude.”
Quote in the paperNot a substantial claimClassified as probably not a substantial claim.
PremisereasoningIn quantum computation, an extra translation step is necessary to encode the second-quantized Hamiltonian in a system of qubits prior to treating the problem.
“In quantum computation, an extra 'translation' step is necessary to encode the second-quantized Hamiltonian in a system of quantum bits, or qubits, prior to treating the problem.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationA study [56] used a superconducting processor (transmon qubits) with 2 qubits, a hardware-efficient ansatz, and SPSA to compute the ground state energy of H2.
“| Superconducting processor (transmon qubits) | H 2 | 2 | 'Hardware- efficient' ansatz | SPSA | Ground state energy | 56 |”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [56] source
Not checked in the free run.
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PremisecitationThe dual plane-wave basis set has been proposed as an alternative for electronic structure calculations on quantum computers.
“Recently, the dual plane-wave basis set (dual PW), corresponding to the Fourier transform of PW, has been proposed as an alternative for electronic structure calculations on quantum computers 301”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [301] source
Not checked in the free run.
May combine two claims.
PremisecitationVQE has been experimentally implemented for molecular systems beyond hydrogen, demonstrating the utility and potential of the algorithm for applications in quantum chemistry even on early quantum computers.
“In particular, VQE was experimentally implemented for molecular systems beyond hydrogen as highlighted in Section 5.1, demonstrating the utility and potential of the algorithm for applications in quantum chemistry even on early quantum computers.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites Section 5.1
Not checked in the free run.
May combine two claims.
InterpretationreasoningWhile progress in quantum hardware development carries us toward the utility horizon, progress in quantum algorithm development moves this landmark itself closer into our reach.
“So, while progress in quantum hardware development carries us toward the utility horizon, progress in quantum algorithm development moves this landmark itself closer into our reach.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
PremisereasoningThe description of core electrons might require a large number of PW basis sets due to large energies, and thus PW are usually employed in combination with pseudopotentials to describe core electrons.
“Correspondingly, the description of core electrons might require a large number of PW basis sets due the large energies, and thus, PW are usually employed in combination with pseudopotentials to describe core electrons.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Premisenone givenLarger-weight terms in the Hamiltonian require longer circuits for their simulation.
“This is important for quantum simulation, as largerweight terms in the Hamiltonian require longer circuits for their simulation.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Premisenone givenIn metals, valence electrons have a behavior more similar to free electrons.
“For example, in metals, valence electrons have a behavior more similar to free electrons.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Premisenone givenChoosing basis set functions with the right mathematical properties can significantly facilitate the evaluation of the Hamiltonian elements and expectation values.
“Choosing basis set functions with the right mathematical properties can significantly facilitate the evaluation of the Hamiltonian elements and expectation values.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Premisenone givenThese one-electron functions, also called spin-orbitals, are usually expanded as linear combinations of a set of standard system-independent functions, which are called the basis set functions.
“Furthermore, these one-electron functions, also called spin-orbitals, are usually expanded as linear combinations of a set of standard system-independent functions, which are called the basis set functions .”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Premisenone givenThe variational quantum eigensolver algorithm can be viewed as a quantum version of Ritz's variational method.
“For example, the variational quantum eigensolver algorithm can be viewed as a quantum version of Ritz's variational method.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Premisenone givenVQE is a hybrid quantum-classical algorithm that estimates molecular properties, often the ground state energies, of quantum systems using the variational principle.
“As detailed in Section 5.1, the variational quantum eigensolver (VQE) is a hybrid quantumclassical algorithm that estimates molecular properties, often the ground state energies, of quantum systems using the variational principle.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Premisenone givenGaussian type orbitals have convenient analytical properties that allow efficient schemes for the computational evaluation of molecular integrals.
“GTOs have convenient analytical properties that allow for efficient schemes for the computational evaluation of molecular integrals.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Premisenone givenTight functions (GTOs with large orbital exponents) can be included when the description of core electrons plays an important role in accuracy.
“Similarly, tight functions (GTOs with large orbitals exponents) can be included when the description of core electrons plays an important role in the accuracy.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Premisenone givenIn practice, the set of one-particle functions is finite, introducing an approximation error associated with a truncated representation of the Hilbert space, or basis-set incompleteness.
“In practice, the set of one-particle functions is finite, introducing an approximation error associated with a truncated representation of the Hilbert space, or basis-set incompleteness.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Premisenone givenAtomic basis sets are well suited to describe electron densities in isolated molecules, where the densities resemble those of individual atoms.
“Atomic basis sets are well suited to describe electron densities in isolated molecules, where the densities resemble those of individual atoms.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationThe Bravyi-Kitaev tree method also achieves a mapping with O(log N)-weight creation and annihilation operators.
“A closely related variant of the Bravyi-Kitaev transformation, known as the Bravyi-Kitaev tree method, also achieves a mapping with O (log N )-weight creation and annihilation operators 402 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [402] source
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Premisenone givenThe quality of the dual PW/Sinc basis set can be improved by decreasing the separation in the grid.
“The quality of the basis set can be improved by decreasing the separation in the grid.”
Quote in the paperNot a substantial claimMay combine two claims.
PremisereasoningThe calculation of molecular integrals using STOs has to be carried out numerically because of the lack of analytical solutions, restricting the use of STOs to small molecules.
“Unfortunately, the calculation of the molecular integrals using STOs has to be carried out numerically because of the lack of analytical solutions, restricting the use of STOs to small molecules.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Premisenone givenThe size of the PW basis set is determined by the chosen maximum energy and the volume of the unit cell, and does not depend on the number of atoms within the cell.
“The size of the PW basis set is determined by the chosen value of the maximum energy and the volume V of the unit cell, and in contrast with atomic basis sets, does not depend on the number of atoms within the cell.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Premisenone givenWhen applying quantum computation for chemistry, the Hamiltonian is often considered in the second-quantized formulation, in which the system can be described approximately using a finite basis.
“When applying quantum computation for chemistry, the Hamiltonian is instead often considered in the second-quantized formulation, in which the system can be described approximately using a finite basis.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationContracted GTOs have been adopted as the main basis set functions for electronic structure calculations in isolated molecules.
“Contracted GTOs have been adopted as the main basis set functions for electronic structure calculations in isolated molecules 399”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [399] source
Not checked in the free run.
May combine two claims.
Premisenone givenIn the parity mapping, parity operators are low-weight, while occupation operators become high-weight.
“Here, the parity operators are low-weight, while the occupation operators become high-weight.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Premisenone givenSlater-type orbitals capture the appropriate behavior of electron density near and far from the nuclei, with appropriate cusps for s orbitals at the nuclei and exponential decaying tails.
“The main advantage of STOs is that they capture the appropriate behavior of electron density near and far from the nuclei, with appropriate cusps for s orbitals at the nuclei and exponential decaying tails.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Premisenone givenThe mapping step is a general prerequisite for implementing Hamiltonian simulations, the quantum phase estimation algorithm, or the variational quantum eigensolver.
“We note that this step is a general prerequisite for implementing Hamiltonian simulations, the quantum phase estimation algorithm, or the variational quantum eigensolver.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S4 (based). May combine two claims.
Premisenone givenPWs and their dual diagonalize the kinetic and potential operators, respectively.
“PWs and their dual diagonalize the kinetic and potential operators, respectively.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Premisenone givenThe existing basis sets in computational chemistry offer different levels of compromise among these desired qualities.
“The existing basis sets in computational chemistry offer different levels of compromise among these desired qualities.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S4 (based).
Premisenone givenThe mapping from creation and annihilation operators to qubit operators preserves the algebraic relationships (fermionic canonical commutation relations).
“This is achieved by applying a mapping from the set of creation and annihilation operators in the Hamiltonian to the set of operators on qubits, in which the mapping preserves the algebraic relationships, i.e the fermionic canonical commutation relations.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Premisenone givenThe parity mapping is a valid transformation from fermionic operators to qubits.
“While the parity mapping is a valid transformation from fermionic operators to qubits”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S1 (asserted), S3 (load-bearing).
PremisecitationThe most optimistic estimates predict that quantum computers will solve useful problems in 2 to 5 years.
“However, the most optimistic estimates predict that quantum computers will solve useful problems in 2 to 5 years 395 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [395] Barkoutsos, P. K.; Moll, N.; Staar, P. W. J.; Mueller, P.; Fuhrer, A.; Filipp, S.; Troyer, M.; Tavernelli, I. Fermionic Hamiltonians for Quantum Simulations: A General Reduction Scheme. 2017, arXiv:1706.03637 [quant-ph]; https://arxiv.org/abs/1706.03637.
Not checked in the free run.
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Premisenone givenFor the H2 example, the UCC level of theory is equivalent with the exact solution.
“We note that for the H 2 example, this level of theory is equivalent with the exact solution.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Premisenone givenThe performance of a basis set is usually measured as the difference with the undiscretized problem, which would correspond with a hypothetical infinite basis set.
“Correspondingly, the performance of the basis set is usually measured as the difference with the undiscretized problem, which would correspond with a hypothetical infinite basis set.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisereasoningThe development of software platforms such as Rigetti's pyQuil, IBM's QISKit, ETH Zürich's ProjectQ, Google's Cirq, and Xanadu's Strawberry Fields has amplified the synergy between quantum algorithm developers and quantum machines.
“Fortunately, this synergy has been amplified by the development of software platforms such as Rigetti's pyQuil 389 , IBM's QISKit 390 , ETH Z¨ urich's ProjectQ 391 , Google's Cirq 392 , and Xanadu's Strawberry Fields 393 .”
Quote in the paperNot a substantial claim0 of 3 sources checkedProbably not a substantial claim: may not pass S3 (load-bearing).
Premisenone givenVQE is a near-term alternative to the quantum phase estimation algorithm due to its low coherence time requirements.
“While VQE is a near-term alternative to the quantum phase estimation algorithm due to its low coherence time requirements”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Premisenone givenThe electronic system as written in the first-quantized picture assumes an infinite-dimensional Hilbert space.
“The electronic system, as written in the first-quantized picture in Equation 100, assumes an infinite-dimensional Hilbert space.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Premisenone givenDiffuse functions (GTOs with small exponents) can be included to improve the description of tails of the density, especially in systems such as anions.
“In addition to polarization functions, diffuse functions (GTOs with small exponents) can be included to improve the description of tails of the density, especially in system such as anions.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationMore conservative estimates predict that the quantum utility crossover will occur in 15 to 20 years.
“The more conservative estimates predict that this crossover will occur in 15 to 20 years 394 .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [394] source
Not checked in the free run.
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Premisenone givenUnlike STOs, GTOs do not describe the cusp and exponential tails of the electronic density correctly.
“However, unlike STOs, they do not describe the cusp and exponential tails of the electronic density correctly.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).
InterpretationreasoningThis points to the increasing value in having quantum algorithm developers working closely with quantum machines.
“This points to the increasing value in having quantum algorithm developers working closely with quantum machines.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationA study [56] used a superconducting processor (transmon qubits) with 4 qubits, a hardware-efficient ansatz, and SPSA to compute the ground state energy of LiH.
“| | LiH | 4 | 'Hardware- efficient' ansatz | SPSA | Ground state energy | 56 |”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [56] source
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
PremisecitationA study [316] used a silicon photonic chip with 2 qubits, a 'Parametrized Hamiltonian' ansatz with truncation scheme, and PSO to compute ground and excited state energies of two chlorophyll units in an 18-mer ring of the LHII complex.
“| Silicon photonic chip | Two chlorophyll units in 18-mer ring of LHII complex | 2 | 'Parametrized Hamiltonian' ansatz with truncation scheme | PSO | Ground and excited state energies | 316 |”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [316] source
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
PremisecitationA study [56] used a superconducting processor (transmon qubits) with 6 qubits, a hardware-efficient ansatz, and SPSA to compute the ground state energy of BeH2.
“| | BeH 2 | 6 | 'Hardware- efficient' ansatz | SPSA | Ground state energy | 56 |”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [56] source
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationA study [59] used an ion trap processor (Ca+ ions) with 2 qubits, UCC, and grid scan with local optimization to compute the ground state energy of H2.
“| Ion trap processor (Ca + ions) | H 2 | 2 | UCC | Grid scan and locally optimize | Ground state energy | 59 |”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [59] source
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
PremisecitationA study [58] used a superconducting processor (transmon qubits) with 2 qubits, a hardware-specific parametrized ansatz, and PSO to compute ground and excited state energies of H2.
“| Superconducting processor (transmon qubits) | H 2 | 2 | Hardware- specific parametrized ansatz | PSO | Ground and excited state energies | 58 |”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [58] Poulin D., 2014, Quantum Inf. Comput.
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PremisecitationA study [59] used an ion trap processor (Ca+ ions) with 3 qubits, approximate UCC, and grid scan with local optimization to compute the ground state energy of LiH.
“| | LiH | 3 | Approximate UCC | Grid scan and locally optimize | Ground state energy | 59 |”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [59] source
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Probably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationA study [330] used a superconducting processor (transmon qubits) via cloud with 2-3 qubits, UCC, and grid scan to compute the ground state energy of deuteron.
“| Superconducting processor (transmon qubits) via Cloud | Deuteron | 2-3 | UCC | Grid scan | Ground state energy | 330 |”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [330] source
Not checked in the free run.
The authors state this as an assumption.
There is always a gap between the ground state of H(s) and the first excited state during the adiabatic evolution.
“Assuming there is always a gap between the ground state of H ( s ) and the first excited state, the end of the adiabatic evolution of the state of the system should be close to the ground state of H 1 .”
The authors state this as an assumption.
The Born-Oppenheimer approximation treats nuclei as stationary point charges because nuclear mass is three orders of magnitude greater than electron mass.
“Because the mass of a nucleus is three orders of magnitude greater than that of an electron, a common simplifying assumption made in quantum chemistry calculations, called the Born-Oppenheimer approximation (BOA), treats the nuclei as stationary point charges.”
The authors state this as an assumption.
In most applications of time-dependent DFT, the time-dependent part of the Hamiltonian is assumed to be small.
“Even though the original formulation of Time-Dependent DFT 158 does not require the time-dependent part of the Hamiltonian to be small, it has mostly been applied using this assumption 159-161 .”
The authors state this as an assumption.
In early Hamiltonian simulation works, the Hamiltonian H can be written as a sum of local terms H = ∑_j H_j where each H_j acts on a subsystem.
“Early works on Hamiltonian simulation 37,45,186 assume that H can be written as a sum of local terms H = ∑ j H j where each H j acts on a subsystem.”
The authors state this as an assumption.
In the oracle model for Hamiltonian simulation, the Hamiltonian is a sparse matrix and its elements are provided by a black box or oracle.
“An alternative setting of Hamiltonian simulation 187,188 considers an 'oracle model' where the Hamiltonian is assumed to be a sparse matrix and its elements are provided by a black box (or oracle )”
The authors state this as an assumption.
In the oracle model, one can make queries to the oracle in superposition and obtain answers in superposition.
“It is also assumed that one can make queries to the oracle in superposition and obtain answers in superposition.”
The authors state this as an assumption.
In algorithms for simulating open quantum systems, the Hamiltonian and Lindblad operators are sparse and given by oracles or black boxes.
“These algorithms focus on the Hamiltonian and Lindblad operators being sparse and given by oracles (black-boxes).”
The authors state this as an assumption.
The number of terms ℓ in the Hamiltonian scales polynomially with the system size N.
“Assuming that the number of terms ℓ in the Hamiltonian scales polynomially with the system size N (such as the number of particles or the number of qubits)”
The authors state this as an assumption.
The underlying quantum computer is fault tolerant for the quantum algorithms discussed in Section 4.
“here we focus on quantum algorithms which assume the underlying quantum computer is fault tolerant.”
The paper relies on this without support.
The amplitudes of a wave function stored in quantum bits cannot be efficiently accessed.
“The caveat is that the amplitudes of a wave function stored in quantum bits cannot be efficiently accessed.”
The paper relies on this without support.
Creating a thermal state is computationally difficult, but an approximation may suffice.
“While creating a thermal state is a computationally difficult problem by itself, an approximation may suffice.”
The paper relies on this without support.
The representation of the Hamiltonian in the basis of Slater determinants (the configuration interaction matrix) is sparse.
“The representation of the Hamiltonian in the basis of Slater determinants (the configuration interaction matrix) is sparse and therefore techniques from Section 4.1 can be used for simulation.”
The authors state this as an assumption.
In adiabatic state preparation, the transition between the initial Hamiltonian H0 and the target Hamiltonian H1 is continuous.
“This process assumes that the transition between H 0 and H 1 is continuous and that the ground state energy of λH 0 +(1 -λ ) H 1 remains sufficiently separated from the first excited state energy during the preparation”
The authors state this as an assumption.
In adiabatic state preparation, the ground state energy of λH0+(1-λ)H1 remains sufficiently separated from the first excited state energy during the preparation.
“This process assumes that the transition between H 0 and H 1 is continuous and that the ground state energy of λH 0 +(1 -λ ) H 1 remains sufficiently separated from the first excited state energy during the preparation”
The paper relies on this without support.
Reaction rates are exponentially sensitive to changes in energy; a change of only 1.4 kcal/mol in the free energy generates a change of one order of magnitude in the estimate of a reaction rate.
“Reaction rates, for example, are exponentially sensitive to changes in energy: a change of only 1.4 kcal/mol in the free fenergy (which is proportional to the energy) generates a change of one order of magnitude in the estimate of a reaction rate.”
The authors state this as an assumption.
All Hamiltonian terms are measured independently, i.e. Cov(〈O_i〉, 〈O_j〉) = 0 for all i ≠ j.
“Assuming that all the terms are measured independently i.e Cov( 〈 O i 〉 , 〈 O j 〉 ) = 0 ∀ i = j ), the total variance of the estimator of 〈 H 〉 , will be given by the sum of the variances of the individual Hamiltonian terms”
The authors state this as an assumption.
Parallel rotation and orbital readout is available for Hamiltonian averaging.
“Hamiltonian averaging only adds a constant factor to the state preparation procedure, assuming parallel”
The authors state this as an assumption.
The bound on the number of measurements for the plane wave dual basis assumes a neutral system and a fixed density.
“This bound is obtained by bounding the norms of the different terms in the Hamiltonian assuming a neutral system and a fixed density.”
The paper relies on this without support.
In finite molecules and bulk materials, the number of electrons η is in O(N).
“Since in finite molecules and bulk materials, η ∈ O ( N ), this shows that the number of measurements does not scale worse than O ( N 4 /ϵ 2 ) for this representation of the Hamiltonian.”
The authors state this as an assumption.
In AAVQE, the gap of H(s) does not become too small.
“As long as the gap of H ( s ) does not become too small (see Section 5.3.1), the ground state of H ( s ) will lie close to the ground state of the Hamiltonian H ( s +∆ s ) in the subsequent step.”
The paper relies on this without support.
Measurement noise has a significant impact on VQE-based calculations, especially when iterative procedures are involved.
“In all these calculations, an important factor to consider is the impact of measurement noise, especially when iterative procedures are involved.”
The authors state this as an assumption.
The Born-Oppenheimer approximation, treating the nuclei as stationary classical particles, is valid for the molecular hydrogen problem.
“To simplify the problem, we apply the Born-Oppenheimer approximation, in which we treat the nuclei as stationary classical particles.”
The paper relies on this without support.
In periodic systems, the periodic density does not vanish exponentially far away from the nuclei.
“In periodic systems, the periodic density does not vanish exponentially far away from the nuclei, unlike the case of isolated molecules.”
The paper relies on this without support.
In isolated molecules, the electronic density is highly peaked around the nuclei and vanishes exponentially away from them.
“In isolated molecules, the electronic density is highly peaked around the nuclei and vanishes exponentially away from them”
The paper relies on this without support.
Compared to GTOs, PW basis sets generally require an order of magnitude more functions to achieve a similar accuracy.
“Compared to GTOs, PW basis sets generally require an order of magnitude more functions to achieve a similar accuracy.”
The paper relies on this without support.
So far, the parity mapping has not been considered as a useful contender for use in quantum simulation.
“While the parity mapping is a valid transformation from fermionic operators to qubits, so far, it has not been considered as a useful contender for use in quantum simulation.”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
Quantum computers can play a crucial role in performing precise simulations for chemical processes and procedures in biological sciences.
“Therefore, quantum computers can play a crucial role in performing precise simulations for chemical processes and procedures in biological sciences [9, 69-71].”
From Quantum Computing in the Next-Generation Computational Biology Landscape: From Protein Folding to Molecular Dynamics (2024) · cites this paper as [9]
The electronic structure problem is one of the most prominent tasks within VQAs and was the pioneering task for VQE.
“The electronic structure problem is one of the most prominent tasks within VQAs (see for example the reviews (Cao et al. , 2019; McArdle et al. , 2020)) and was the pioneering task for VQE (McClean et al. , 2016; Peruzzo et al. , 2014).”
From Noisy intermediate-scale quantum algorithms (2022) · cites this paper as Cao et al. , 2019
Quantum computers have been proposed to simulate quantum systems efficiently.
“By exploiting their quantum mechanical properties, quantum computers have been proposed to simulate quantum systems efficiently [9–13].”
From Drug design on quantum computers (2024) · cites this paper as [10]
Preparation of a generic input state requires sophisticated techniques such as quantum random access memory, leading to a data input problem.
“Second, the preparation of a generic input state |b requires sophisticated techniques such as quantum random access memory (QRAM) [35], thus leading to a so-called data”
From Solving nonlinear differential equations with differentiable quantum circuits (2021) · cites this paper as [35]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
The stated creation action uses a matrix that annihilates the unoccupied state
MinorThe paper says σ+ performs the occupation-changing action of a creation operator, but its displayed matrix sends |0〉 to zero, not to |1〉. This conflicts with its stated convention that occupation is represented by 1.
“This is a consequence of the creation operator carrying out two actions: the action of changing the occupation is carried out locally by σ + i”
“where σ z = [ 1 0 0 -1 ] and σ + = [ 0 1 0 0 ] .”
Pauli-operator indices are defined in opposite orders
MinorThe text defines σ i,j as the j-th operator in the i-th Hamiltonian term, but the glossary calls it the i-th operator acting on the j-th qubit. These assign different meanings to the indices.
“In Equation 15, the notation σ i,j means the j -th operator in the i -th term of the Hamiltonian.”
“| σ i,j | i -th Pauli operator acting on the j -th qubit |”
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“Therefore, quantum computers can play a crucial role in performing precise simulations for chemical processes and procedures in biological sciences [9, 69–71].”
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Orbital-optimized pair-correlated electron simulations on trapped-ion quantum computers
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Towards practical applications in quantum computational biology
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Quantum computational quantification of protein–ligand interactions
Kirsopp, J. J. M., Di Paola, C., Manrique, D. Z., Krompiec, M., Greene‐Diniz, G., Guba, W., Meyder, A., Wolf, D., Strahm, M., & Muñoz Ramo, D. (2022) · International Journal of Quantum Chemistry · doi:10.1002/qua.26975 · Cited by 59 · see citing works
RetractedMutual information-assisted adaptive variational quantum eigensolver
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Self-citationApplication of Quantum Computing to Biochemical Systems: A Look to the Future
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Benchmarking Adaptive Variational Quantum Eigensolvers
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Quantum computing algorithms: getting closer to critical problems in computational biology
Marchetti, L., Nifosì, R., Martelli, P. L., Da Pozzo, E., Cappello, V., Banterle, F., Trincavelli, M. L., Martini, C., & D’Elia, M. (2022) · Briefings in Bioinformatics · doi:10.1093/bib/bbac437 · Cited by 47 · see citing works
“Detailed accounts of quantum chemistry in the context of quantum computation may also be found in other recent reports [86, 89, 92].”
ResultTowards the simulation of large scale protein–ligand interactions on NISQ-era quantum computers
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Polymer physics by quantum computing
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Quantum-Train Long Short-Term Memory: Application on Flood Prediction Problem
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RNA folding using quantum computers
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Castaldo, D., Rosa, M., & Corni, S. (2020) · arXiv · doi:10.48550/arxiv.2007.00368 · Cited by 32
The landscape of the quantum start-up ecosystem
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Error mitigation with Clifford quantum-circuit data
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Dominant Reaction Pathways by Quantum Computing
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Quantum computation of eigenvalues within target intervals
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Self-citationQuantum Computing in Civil Engineering: Potentials and Limitations
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Limitations of Quantum Hardware for Molecular Energy Estimation Using VQE
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Quantum error correction and fault tolerance: A comprehensive tutorial
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- Authors
- Cao, Y., Romero, J., Olson, J. P., Degroote, M., Johnson, P. D., Kieferová, M., Kivlichan, I. D., Menke, T., Peropadre, B., Sawaya, N. P. D., Sim, S., Veis, L., & Aspuru-Guzik, A.
- Affiliations
- Harvard University (US)
- University of Waterloo (CA)
- Zapata (United States) (US)
- Intel (United States) (US)
- Czech Academy of Sciences (CZ)
- Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138, United States
- ORCID
- Venue
- Chemical Reviews
- Publisher
- American Chemical Society (ACS)
- Published
- 2019-08-30
- Volume
- 119
- Issue
- 19
- Pages
- 10856-10915
- ISSN
- 0009-2665 (ISSN-L)
- 1520-6890
- OpenAlex
- W2906538035
- Semantic Scholar
- 1eaab9b33f1261744567455a14830e8a92796cf5
- Cited by
- 1,560
- References
- 396
- Citations by year
- 2026: 233 · 2025: 316 · 2024: 275 · 2023: 272 · 2022: 213 · 2021: 164 · 2020: 77 · 2019: 9
- FWCI
- 7.53
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- Locations
- Chemical Reviews (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Government of Canada
- Army Research Office (W911NF-15-1-0256)
- Grantov? Agentura Cesk? Republiky (18-18940Y)
- Intelligence Advanced Research Projects Activity (W911NF-17-C-0050)
- Office of Naval Research (ONR 00014-16-1-2008)
- U.S. Department of Energy (DE-FG02-97ER25308)
- Natural Sciences and Engineering Research Council of Canada
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Advanced Chemical Physics Studies
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Keywords
- quantum computing, quantum chemistry, quantum simulation, quantum algorithms, molecular electronic structure, quantum computing hardware, superposition, entanglement
- MeSH terms
- Algorithms, Computing Methodologies, Models, Chemical, Quantum Theory, Molecular Dynamics Simulation
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
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