The DOI matches its Crossref record; 396 of the 396 references it cites were found in a registry or an index; 3 of the 446 sources its claims cite were checked for support.
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.
Overview
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Written from the paper’s full text.
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
292 Not checked yet · 173 No basis given · 86 Own interpretation · 1 Partly supported · 1 Couldn’t confirm · 1 Own result · accuracy not yet measured
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In the paper
“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 :”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.3.1 Adiabatic · p. 120
What the sources say
- Could not verifycites [181] sourceExperimental — accuracy not yet measured
The passages in “Variational Ansatz for excitations” discuss a Heisenberg spin model, while “Topologically trivial states” discusses a general lattice Hamiltonian; neither gives a molecular-hydrogen Hamiltonian. Searches for molecular hydrogen in a minimal basis and for the Bravyi–Kitaev transformation found no passage addressing the claimed four-qubit mapping. Page numbers were unavailable in our copy. (LLM, conf=0.75)
- Partly supportedcites [366] sourceExperimental — accuracy not yet measured
Section 6.4.1 reports obtaining a diatomic-hydrogen Hamiltonian in the STO-3G basis. The passages from our copy do not give a four-qubit Bravyi–Kitaev mapping or the specific Hamiltonian form attributed to this source. (LLM, conf=0.83)
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In the paper
“Using such recipes, implementing e -iHt for an n -qubit Hamiltonian takes time polynomial in n on a quantum computer”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3.2 Dynamics: Hamiltonian simulation and hybrid methods · p. 32
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3.1 Statics: phase estimation and variational quantum eigensolvers · p. 30
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In the paper
“Applying phase estimation on the initial state | φ 〉| 0 〉 yields ∑ j β j | ψ j 〉| ˜ ϕ j 〉 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3.1 Statics: phase estimation and variational quantum eigensolvers · p. 30
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In the paper
“Initial simulations show that this method is competitive with the traditional gradient descent approach of the variational quantum eigensolver.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.2.2 Imaginary-time variational quantum simulator · p. 119
What the sources say
- Not checked yetcites [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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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > C.5 Variational quantum eigensolver for quantum chemistry · p. 149
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- Also stated in: § C.5 Variational quantum eigensolver for quantum chemistry · p. 150 · plain
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 1 Introduction and historical overview · p. 8
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In the paper
“A quantum computer, as envisioned by Feynman and Manin, can potentially circumvent the roadblock of exponential cost by being quantum mechanical itself.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 1 Introduction and historical overview · p. 7
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 1 Introduction and historical overview · p. 5
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 1 Introduction and historical overview · p. 5
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- Probably not a substantial claim. 232
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The cited sources that could be checked couldn’t confirm or dispute this claim.
In the paper
“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”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > A Quantum chemistry basis sets · p. 137
What the sources say
- Could not verifycites [301] sourceExperimental — accuracy not yet measured
The Abstract (p. 1) describes a semiclassical Fourier transform for Shor’s algorithms, not a basis set for electronic-structure calculations. Searches for “dual plane-wave basis set” and “plane wave basis electronic structure” found no passage in our copy addressing the claimed proposal. (LLM, conf=0.78)
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 1 Introduction and historical overview · p. 8
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 1 Introduction and historical overview · p. 7
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 1 Introduction and historical overview · p. 8
What the sources say
- Not checked yetcites [37] source
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 1 Introduction and historical overview · p. 8
What the sources say
- Not checked yetcites [45] source
- Not checked yetcites [46] Manin Y., 1980
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 1 Introduction and historical overview · p. 7
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In the paper
“Around the same time, Wiesner 40 and Zalka 41 also suggested using quantum computers for simulating the dynamics of many-body systems.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 1 Introduction and historical overview · p. 8
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.1 Static properties: wave functions and energy spectra · p. 18
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In the paper
“Coupled cluster 79 is formulated as a compact wave function parametrization in terms of an exponential functional of a cluster operator.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.1 Static properties: wave functions and energy spectra · p. 16
What the sources say
- Not checked yetcites [79] source
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.1 Static properties: wave functions and energy spectra · p. 18
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.1 Basics and challenges of classical quantum chemistry · p. 15
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- Also stated in: § 2.1 Basics and challenges of classical quantum chemistry · p. 15 · plain
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.1 Basics and challenges of classical quantum chemistry · p. 14
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.1 Basics and challenges of classical quantum chemistry · p. 14
What the sources say
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 1 Introduction and historical overview · p. 9
What the sources say
- Not checked yetcites [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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In the paper
“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).”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.1 Static properties: wave functions and energy spectra · p. 18
What the sources say
- Not checked yetcites [79] source
- Not checked yetcites [92] Helgaker T., 2000 source
- Not checked yetcites [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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In the paper
“In contrast, the Boson Sampling model of computation does not have this property 63 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 1 Introduction and historical overview · p. 10
What the sources say
- Not checked yetcites [63] source
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In the paper
“This has inspired formulations of various physically-motivated wave function approximations as well as quantum chemistry algorithms on classical computers.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > Quantum chemistry in the age of quantum computing · p. 13
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 1 Introduction and historical overview · p. 10
What the sources say
- Not checked yetcites [62] source
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In the paper
“However, the quality of DFT depends on the chosen exchange-correlation functional.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.1 Basics and challenges of classical quantum chemistry · p. 14
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“It has been shown that quantum computers avoid the dynamical sign problem 43 providing a more effective alternative to classical methods for quantum simulation.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.1 Static properties: wave functions and energy spectra · p. 18
What the sources say
- Not checked yetcites [43] source
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.1 Basics and challenges of classical quantum chemistry · p. 14
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In the paper
“However, it does give rise to a class of sampling problems 64 which are hard to solve on a classical computer.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 1 Introduction and historical overview · p. 10
What the sources say
- Not checked yetcites [64] source
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In the paper
“A diverse set of experimental groups worldwide have already implemented various variational quantum algorithms on their physical systems 52-59 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 1 Introduction and historical overview · p. 9
What the sources say
- Not checked yetcites [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.
- Not checked yetcites [53] source
- Not checked yetcites [54] Kitaev A., 2002 source
- Not checked yetcites [55] source
- Not checked yetcites [56] source
- Not checked yetcites [57] Hastings M. B., 2015, Quantum Inf. Comput.
- Not checked yetcites [58] Poulin D., 2014, Quantum Inf. Comput.
- Not checked yetcites [59] source
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“QMC relies on the stochastic estimation of the energy of a trial wave function”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.1 Static properties: wave functions and energy spectra · p. 17
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.1 Static properties: wave functions and energy spectra · p. 18
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In the paper
“Exact simulation of quantum chemistry systems is widely regarded as one of the problems that would benefit enormously from quantum hardware 29 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.1 Static properties: wave functions and energy spectra · p. 18
What the sources say
- Not checked yetcites [29] source
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In the paper
“Correlation effects are thereby completely neglected, meaning computational approximations for various physical quantities are usually of low quality.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.1 Basics and challenges of classical quantum chemistry · p. 15
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.1 Static properties: wave functions and energy spectra · p. 17
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In the paper
“It has been very successful in performing large-scale calculations for extended systems in quantum chemistry.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.1 Static properties: wave functions and energy spectra · p. 17
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 1 Introduction and historical overview · p. 9
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.1 Basics and challenges of classical quantum chemistry · p. 15
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In the paper
“The quantum computational cost of these algorithms has continued to be reduced 47-50 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 1 Introduction and historical overview · p. 8
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In the paper
“Despite all these advances, FCI is still the most useful as a benchmarking method for less-costly quantum chemical methods.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.1 Static properties: wave functions and energy spectra · p. 18
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In the paper
“These characterizations, discussed in Chapter 3, provide further motivation for the development of quantum algorithms targeted at solving them.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 1 Introduction and historical overview · p. 10
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3 Going beyond classical limitations using quantum computers · p. 28
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In the paper
“It is sufficient to use classical MD methods which simulate the problem using an effective force field and Newtonian mechanics 140 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.2 Dynamical properties: time evolution of the wave function · p. 22
What the sources say
- Not checked yetcites [140] source
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3 Going beyond classical limitations using quantum computers · p. 27
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3 Going beyond classical limitations using quantum computers · p. 26
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.2 Dynamical properties: time evolution of the wave function · p. 26
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- 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.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.2 Dynamical properties: time evolution of the wave function · p. 22
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.1 Static properties: wave functions and energy spectra · p. 19
What the sources say
- Not checked yetcites [110] source
plaindirect- Asserted: passes — The authors put it forward as true.
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- Also stated in: § 2.2.1 Static properties: wave functions and energy spectra · p. 19 · plain
In the paper
“Matrix product states allow for a compact representation of one-dimensional systems and also satisfy an area law 111 in one dimension.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.1 Static properties: wave functions and energy spectra · p. 19
What the sources say
- Not checked yetcites [111] Harrison R., 1994
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.2 Dynamical properties: time evolution of the wave function · p. 23
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.2 Dynamical properties: time evolution of the wave function · p. 23
What the sources say
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.1 Static properties: wave functions and energy spectra · p. 19
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.2 Dynamical properties: time evolution of the wave function · p. 26
What the sources say
- Not checked yetcites [175] source
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3 Going beyond classical limitations using quantum computers · p. 28
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.2 Dynamical properties: time evolution of the wave function · p. 24
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3 Going beyond classical limitations using quantum computers · p. 27
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In the paper
“They can be adapted to perform quantum propagation 150,152 , although they are not the most computationally efficient choice for a given accuracy.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.2 Dynamical properties: time evolution of the wave function · p. 24
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In the paper
“The time-dependent variational principle (TDVP) 167-169 has enabled time-dependent variational Monte Carlo calculations for bosons 170 and strongly correlated fermions 171 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.2 Dynamical properties: time evolution of the wave function · p. 25
What the sources say
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.2 Dynamical properties: time evolution of the wave function · p. 24
What the sources say
- Not checked yetcites [153] Leach A., 2001
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“Such effects include nuclear tunneling and zero-point vibrational energy, which are not captured by Newtonian mechanics.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.2 Dynamical properties: time evolution of the wave function · p. 23
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In the paper
“In fact, there are promising ideas which may bring tangible results for quantum chemistry on near-term devices.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3 Going beyond classical limitations using quantum computers · p. 27
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.2 Dynamical properties: time evolution of the wave function · p. 22
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3 Going beyond classical limitations using quantum computers · p. 26
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3 Going beyond classical limitations using quantum computers · p. 27
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.2 Dynamical properties: time evolution of the wave function · p. 26
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3 Going beyond classical limitations using quantum computers · p. 27
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“However, as we will discuss in this section, quantum computers allow for efficient implementation of these two features.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3 Going beyond classical limitations using quantum computers · p. 26
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.1 Static properties: wave functions and energy spectra · p. 19
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3 Going beyond classical limitations using quantum computers · p. 28
plaindirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Linear response is the first-order expansion of a time-dependent property and perturbation theory falls short of describing the full time dependence.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.2 Dynamical properties: time evolution of the wave function · p. 25
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.2 Dynamical properties: time evolution of the wave function · p. 24
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.2 Dynamical properties: time evolution of the wave function · p. 21
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.2.2 Dynamical properties: time evolution of the wave function · p. 23
What the sources say
- Not checked yetcites [150] source
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 3.1 Complexity classes · p. 37
What the sources say
- Not checked yetcites [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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In the paper
“Naturally, any discussion of realizing the operation e -iHt is restricted to simulating time evolution of closed , time-independent systems.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3.2 Dynamics: Hamiltonian simulation and hybrid methods · p. 33
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3.1 Statics: phase estimation and variational quantum eigensolvers · p. 31
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In the paper
“A salient feature of hybrid quantum-classical algorithms is that much of the computational burden is offloaded to a classical computer.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3.1 Statics: phase estimation and variational quantum eigensolvers · p. 31
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“An important class of hybrid quantum-classical algorithms that are useful for solving static problems in quantum chemistry is the variational quantum eigensolver (VQE)”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3.1 Statics: phase estimation and variational quantum eigensolvers · p. 31
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3.1 Statics: phase estimation and variational quantum eigensolvers · p. 30
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In addition, the Hamiltonians for molecular electronic structure problems are also well studied in the context of implementing time evolution U = e -iHt .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3.1 Statics: phase estimation and variational quantum eigensolvers · p. 30
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Phase estimation is an algorithmic framework that encapsulates a broad set of quantum algorithms.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3.1 Statics: phase estimation and variational quantum eigensolvers · p. 30
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The time evolution of an initial state | ψ ( ⃗ θ 0 ) 〉 then translates to updating the classical parameters iteratively as the simulation progresses.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3.2 Dynamics: Hamiltonian simulation and hybrid methods · p. 33
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 3.1 Complexity classes · p. 37
What the sources say
- Not checked yetcites [203] source
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3.2 Dynamics: Hamiltonian simulation and hybrid methods · p. 32
What the sources say
- Not checked yetcites [43] source
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“For timedependent Hamiltonians, early works 196 focus on variants of Trotter-Suzuki formula, which are subsequently improved with black-box models 188 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3.2 Dynamics: Hamiltonian simulation and hybrid methods · p. 33
What the sources say
plaindirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 3 Computational complexity · p. 35
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“the time step t is small, for which e -iHt ≈ I -iHt . This yields a linear response theory;”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3.2 Dynamics: Hamiltonian simulation and hybrid methods · p. 32
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“By measuring the state | ˜ ϕ j 〉 one could extract the phase ϕ i up to the precision allowed by the qubit register.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3.1 Statics: phase estimation and variational quantum eigensolvers · p. 30
hedgeddirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Although the exact relationships between the complexity classes remain largely open, certain containment relationships have already been proven (Figure 2).”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 3.1 Complexity classes · p. 38
plain- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 3 Computational complexity · p. 36
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“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).”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3.2 Dynamics: Hamiltonian simulation and hybrid methods · p. 33
What the sources say
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
- 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.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“There is a sense in which problems in #P are at least as hard as the corresponding problems in NP .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 3.1 Complexity classes · p. 37
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 3.1 Complexity classes · p. 38
hedgedconditional- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“For quantum chemistry applications, there are well motivated methods for preparing initial wave functions based on efficiently computable ansatzes.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3.1 Statics: phase estimation and variational quantum eigensolvers · p. 30
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In the paper
“Recently, simulation techniques in the interaction picture 197 have been proposed.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3.2 Dynamics: Hamiltonian simulation and hybrid methods · p. 33
What the sources say
- Not checked yetcites [197] source
plain- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Instead they require fault-tolerant quantum computers which can in principle perform quantum computation indefinitely.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3.1 Statics: phase estimation and variational quantum eigensolvers · p. 30
plaindirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3.1 Statics: phase estimation and variational quantum eigensolvers · p. 30
plaindirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Yet, there are heuristic methods which regularly solve the Hartree-Fock problem in practice.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 3 Computational complexity · p. 36
plain- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3.1 Statics: phase estimation and variational quantum eigensolvers · p. 31
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 3.1 Complexity classes · p. 38
What the sources say
- Not checked yetcites [67] source
plaincorrelational- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In these methods, a quantum state | ψ ( ⃗ θ ) 〉 is generated by a low depth circuit with classical parameters ⃗ θ .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3.2 Dynamics: Hamiltonian simulation and hybrid methods · p. 33
plaindirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 3 Computational complexity · p. 36
boostedcorrelational- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4 Quantum simulation algorithms for fault-tolerant quantum computers · p. 48
plain- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In practice, quantum computers are error-prone due to both imprecision of processes and unwanted interaction with the environment.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4 Quantum simulation algorithms for fault-tolerant quantum computers · p. 48
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“However, it is possible to correct the errors and the system of detecting and correcting for errors is called an error correcting code 237 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4 Quantum simulation algorithms for fault-tolerant quantum computers · p. 48
What the sources say
- Not checked yetcites [237] Ryser H. J., 1963 source
hedged- Asserted: passes — The authors put it forward as true.
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In the paper
“finding the ground state of Bose-Hubbard model is QMA -complete 210”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 3.1 Complexity classes · p. 39
What the sources say
- Not checked yetcites [210] Cleve, R.; Wang, C. Efficient Quantum Algorithms for Simulating Lindblad Evolution. 2016, arXiv:1612.09512 [quant-ph]; https://arxiv.org/abs/1612.09512.
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.1.1 General Hamiltonian simulation · p. 54
What the sources say
- Not checked yetcites [250] source
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4 Quantum simulation algorithms for fault-tolerant quantum computers · p. 48
What the sources say
- Not checked yetcites [237] Ryser H. J., 1963 source
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
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None of the sources cited for this claim have been checked yet.
In the paper
“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 ).”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4 Quantum simulation algorithms for fault-tolerant quantum computers · p. 47
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 3.2 Complexity theory of the electronic structure problem · p. 40
hedged- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.1.1 General Hamiltonian simulation · p. 52
What the sources say
- Not checked yetcites [253] source
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
None of the sources cited for this claim have been checked yet.
In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.1.1 General Hamiltonian simulation · p. 50
What the sources say
- Not checked yetcites [37] source
- Not checked yetcites [245] source
- Not checked yetcites [246] source
- Not checked yetcites [247] source
- Not checked yetcites [248] Kitaev, A. Y. Quantum Measurements and the Abelian Stabilizer Problem. 1995, arXiv:quant-ph/9511026; https://arxiv.org/abs/quant-ph/9511026.
- Not checked yetcites [249] source
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“the 3D Ising model is NP -complete 209”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 3.1 Complexity classes · p. 39
What the sources say
- Not checked yetcites [209] source
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In the paper
“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 〉 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.1.1 General Hamiltonian simulation · p. 53
What the sources say
- Not checked yetcites [189] source
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In the paper
“finding the universal functional in density functional theory is also QMA -complete 65 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 3.1 Complexity classes · p. 39
What the sources say
- Not checked yetcites [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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In the paper
“The structure assumed in Lloyd's analysis is that the Hamiltonian describes only local interactions .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.1.1 General Hamiltonian simulation · p. 50
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- Not checked yetcites [37] source
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 3.2 Complexity theory of the electronic structure problem · p. 40
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In the paper
“The advantage of the digital simulation approach is its aim for universality : any feasible Hamiltonian can be digitally simulated.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.1.1 General Hamiltonian simulation · p. 50
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In the paper
“The quantum algorithms that we discuss in this section were designed for quantum computers that function without error.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4 Quantum simulation algorithms for fault-tolerant quantum computers · p. 48
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4 Quantum simulation algorithms for fault-tolerant quantum computers · p. 48
What the sources say
- Not checked yetcites [238] Gard B. T., 2015 source
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4 Quantum simulation algorithms for fault-tolerant quantum computers · p. 48
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In the paper
“Independently, Aharonov and Ta-Shma 253 introduced the sparse Hamiltonian paradigm and thus addressed the first two questions above.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.1.1 General Hamiltonian simulation · p. 52
What the sources say
- Not checked yetcites [253] source
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“The state-of-art quantum algorithms achieve scaling roughly logarithmic in inverse precision and linear in time.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4 Quantum simulation algorithms for fault-tolerant quantum computers · p. 47
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In the paper
“By chance, this can also be the global optimum, but proving that it is, or systematically finding the global minimum, is NP -complete.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 3.2 Complexity theory of the electronic structure problem · p. 40
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 3.2 Complexity theory of the electronic structure problem · p. 39
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In the paper
“Quantum computers are particularly useful when classical methods cannot provide answers with high-accuracy such as for strongly correlated systems.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.1 State preparation · p. 63
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“The ansatz for the preparation comes from a classical approximation for a ground state of a given Hamiltonian.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.1 State preparation · p. 63
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.1 State preparation · p. 63
What the sources say
- Not checked yetcites [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
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.1.3 Quantum Fourier transform · p. 62
What the sources say
- Not checked yetcites [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.
- Not checked yetcites [273] Childs, A. M.; Kothari, R. Simulating Sparse Hamiltonians With Star Decompositions, Conference on Quantum Computation, Communication, and Cryptography , 2010; pp 94–103. source
- Not checked yetcites [176] source
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“For small molecules, HF provides a good approximation for the ground state, for example, 0.77 for FeMoCo 287 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.1 State preparation · p. 64
What the sources say
- Not checked yetcites [287] source
plaincorrelationalSingle case- Asserted: passes — The authors put it forward as true.
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In the paper
“Abrams and Lloyd 38 focused on state preparation in more detail and described concrete initialization procedures.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.1 State preparation · p. 63
What the sources say
- Not checked yetcites [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
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“The unitary e i arccos( H/λ ) can be implemented using a quantum walk operator 189 which requires fewer gates than time evolution.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.1.2 Quantum phase estimation · p. 60
What the sources say
- Not checked yetcites [189] source
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“These states have non-zero overlap with a polynomial number of computational basis states, for which there are efficient methods of preparation 288 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.1 State preparation · p. 64
What the sources say
- Not checked yetcites [288] source
plaincorrelational- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“These are, respectively, a polynomial and an exponential improvement over the previous algorithm.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.1 State preparation · p. 63
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“Low and Wiebe 197 introduced Hamiltonian simulation in the interaction picture.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.1.1 General Hamiltonian simulation · p. 56
What the sources say
- Not checked yetcites [197] source
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The success of an algorithm for determining ground state energy depends on the quality of the state preparation.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.1 State preparation · p. 63
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper may give no reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2 Determining the ground states energies of quantum chemistry Hamiltonians · p. 62
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.1 State preparation · p. 63
What the sources say
- Not checked yetcites [279] source
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“In his pioneering work, Zalka 41 considered discretizing the wave function and initializing simulation through a series of controlled rotations.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.1 State preparation · p. 63
What the sources say
- Not checked yetcites [41] source
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Grover and Rudolph 285 gave a similar algorithm to prepare a state corresponding to an efficiently integrable probability distribution.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.1 State preparation · p. 63
What the sources say
- Not checked yetcites [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.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.1 State preparation · p. 63
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. It may state two separate claims as one.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.1.1 General Hamiltonian simulation · p. 55
What the sources say
- Not checked yetcites [188] source
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“However, this would be a costly procedure in practice.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.1 State preparation · p. 63
hedged- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.2 Hamiltonian simulation for quantum chemistry · p. 67
What the sources say
- Not checked yetcites [301] source
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.3 Measuring properties of many-body systems · p. 70
What the sources say
- Not checked yetcites [45] source
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.2 Hamiltonian simulation for quantum chemistry · p. 69
What the sources say
- Not checked yetcites [281] source
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.2 Hamiltonian simulation for quantum chemistry · p. 69
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“Independently of Lloyd's quantum simulation algorithm 37 , Zalka 273 introduced a quantum algorithm for simulation of single- and many-body quantum systems.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.2 Hamiltonian simulation for quantum chemistry · p. 65
What the sources say
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
None of the sources cited for this claim have been checked yet.
In the paper
“Both results use interaction picture simulations 197 and plane wave basis and achieve O ( N 11 / 3 ) and O ( η log N ) complexities.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.2 Hamiltonian simulation for quantum chemistry · p. 70
What the sources say
- Not checked yetcites [197] source
- Not checked yetcites [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
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 79
What the sources say
- Not checked yetcites [322] source
- Not checked yetcites [323] source
- Not checked yetcites [324] source
- Not checked yetcites [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.
- Not checked yetcites [320] source
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.2 Hamiltonian simulation for quantum chemistry · p. 66
What the sources say
- Not checked yetcites [99] source
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.3 Measuring properties of many-body systems · p. 70
hedged- Asserted: passes — The authors put it forward as true.
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In the paper
“For real state simulations, Kivlichan et al. 76 simulated interacting particles and achieved super-polynomially higher accuracy than previous algorithms.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.2 Hamiltonian simulation for quantum chemistry · p. 69
What the sources say
- Not checked yetcites [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
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.2 Hamiltonian simulation for quantum chemistry · p. 67
What the sources say
- Not checked yetcites [47] source
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.1 State preparation · p. 64
What the sources say
- Not checked yetcites [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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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.1 State preparation · p. 64
What the sources say
- Not checked yetcites [45] source
hedgeddirect- Asserted: passes — The authors put it forward as true.
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In the paper
“Standard quantum chemistry methods can be used to lower-bound this gap 45 even though exactly computing it can be demanding.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.1 State preparation · p. 64
What the sources say
- Not checked yetcites [45] source
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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 ).”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.2 Hamiltonian simulation for quantum chemistry · p. 67
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.3 Measuring properties of many-body systems · p. 73
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“The Green's function for a quantum particle is then approximated by a product of exponentials with potential and kinetic energy in the exponents.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 4.2.2 Hamiltonian simulation for quantum chemistry · p. 65
What the sources say
- Not checked yetcites [273] Childs, A. M.; Kothari, R. Simulating Sparse Hamiltonians With Star Decompositions, Conference on Quantum Computation, Communication, and Cryptography , 2010; pp 94–103. source
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Finally, UCCSD has been demonstrated experimentally for HeH + using a single trapped ion 53 , for H 2 on a superconducting quantum computer 55 ,”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 83
What the sources say
- Not checked yetcites [55] source
boostedSingle case- Asserted: passes — The authors put it forward as true.
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In the paper
“Finally, UCCSD has been demonstrated experimentally for HeH + using a single trapped ion 53 , for H 2 on a superconducting quantum computer 55 ,”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 83
What the sources say
- Not checked yetcites [53] source
boostedSingle case- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“and S is the number of repetitions of the circuit block and can be varied to control the approximation accuracy.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 84
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 82
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“It has been pointed out that analytical gradients require orders-of-magnitude fewer measurements than numerical gradients 329 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 83
What the sources say
- Not checked yetcites [329] source
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 84
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 86
What the sources say
- Not checked yetcites [325] source
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Other versions of Equation 37 can be obtained by mimicking higher-order Trotter-Suzuki formulas.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 84
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 88
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“The Hamiltonian variational approach, proposed by Wecker et al. 47 , defines a strategy for building variational ansatze for arbitrary Hamiltonians.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 84
What the sources say
- Not checked yetcites [47] source
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The Hamiltonian variational approach greatly reduces the number of variational parameters, which eases the optimization.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 88
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper may give no reason why it holds. It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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 ).”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 82
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“If all the terms within a subset were to commute, this could be implemented exactly.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 84
hedgedconditional- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Consequently the corresponding CC operator is unitary and the formulation is variational, which offers an advantage compared to standard CC.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 82
plaindirect- Asserted: passes — The authors put it forward as true.
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None of the sources cited for this claim have been checked yet.
In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 83
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 86
What the sources say
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 85
What the sources say
- Not checked yetcites [216] Frenkel J., 1934
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
In the paper
“Wecker et al. 325 also proposed two different optimization strategies for minimizing the energy of the ansatze defined in Equation 37.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 86
What the sources say
- Not checked yetcites [325] source
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 83
What the sources say
- Not checked yetcites [329] source
hedgeddirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
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In the paper
“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).”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 82
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 83
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 81
boosted- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 80
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 82
plainconditional- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The plain wave dual basis set can be viewed as a discrete variable representation (DVR) and is particularly suited for periodic systems.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 89
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 89
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 97
What the sources say
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 89
boosteddirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 90
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 97
What the sources say
- Not checked yetcites [327] source
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“More recently, Barkoutsos et al. 327 proposed a new set of entangling blocks.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 97
What the sources say
- Not checked yetcites [327] source
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 90
hedgeddirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 89
What the sources say
- Not checked yetcites [301] source
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 96
What the sources say
- Not checked yetcites [326] source
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 94
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
None of the sources cited for this claim have been checked yet.
In the paper
“While the other linear-depth methods have better constant factors, their circuits are less general 302,326,335 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 93
What the sources say
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 94
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In the paper
“These ansatze comprise a series of parametrized quantum circuits originally motivated by the limitations of existing NISQ hardware 56 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 97
What the sources say
- Not checked yetcites [56] source
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 97
What the sources say
- Not checked yetcites [56] source
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 94
What the sources say
- Not checked yetcites [326] source
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In the paper
“A similar idea has been used to design ansatze for preparing non-Gaussian states efficiently on classical computers, with high success 338 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 94
What the sources say
- Not checked yetcites [338] source
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 93
What the sources say
- Not checked yetcites [326] source
- Not checked yetcites [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.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 96
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 93
What the sources say
- Not checked yetcites [337] source
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 96
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 90
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 96
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 90
What the sources say
- Not checked yetcites [326] source
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 96
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In the paper
“In this case, the total number of parameters scales as 3 N ( L + 2).”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 97
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.2 Energy estimation in VQE · p. 102
What the sources say
- Not checked yetcites [346] source
plaindirect- Asserted: passes — The authors put it forward as true.
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- Also stated in: § 5.1.2 Energy estimation in VQE · p. 102 · plain
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In the paper
“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,”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.3 Optimization methods · p. 107
What the sources say
- Not checked yetcites [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.
- Not checked yetcites [216] Frenkel J., 1934
- Not checked yetcites [329] source
plaincorrelational- Asserted: passes — The authors put it forward as true.
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- Also stated in: § 5.1.3 Optimization methods · p. 107 · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.2 Energy estimation in VQE · p. 101
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.3 Optimization methods · p. 108
What the sources say
- Not checked yetcites [329] source
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.2 Energy estimation in VQE · p. 103
What the sources say
- Not checked yetcites [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.
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 99
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.2 Energy estimation in VQE · p. 100
What the sources say
- Not checked yetcites [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.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“As we have reviewed earlier in this section, the choice/design of the ansatz is a crucial component for algorithmic performance.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 98
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.3 Optimization methods · p. 108
What the sources say
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.3 Optimization methods · p. 110
What the sources say
- Not checked yetcites [321] source
plainconditional- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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).”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.2 Energy estimation in VQE · p. 104
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.2 Energy estimation in VQE · p. 101
What the sources say
- Not checked yetcites [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.
plain- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > Strategies for reducing the number of measurements in Hamiltonian averaging. · p. 104
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > Strategies for reducing the number of measurements in Hamiltonian averaging. · p. 105
What the sources say
- Not checked yetcites [216] Frenkel J., 1934
plaindirect- Asserted: passes — The authors put it forward as true.
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- Also stated in: § Strategies for reducing the number of measurements in Hamiltonian averaging. · p. 105 · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 98
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > Strategies for reducing the number of measurements in Hamiltonian averaging. · p. 104
What the sources say
- Not checked yetcites [216] Frenkel J., 1934
- Not checked yetcites [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.
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.3 Optimization methods · p. 108
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“For example, VQE relies on ultimately preparing (and then measuring) an entangled state of many qubits.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 98
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > Strategies for reducing the number of measurements in Hamiltonian averaging. · p. 105
What the sources say
- Not checked yetcites [346] source
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.1 Ansatze for state preparation · p. 98
What the sources say
- Not checked yetcites [345] source
plaindirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“However, not all the Hamiltonian terms can be measured simultaneously, which increases the number of measurements needed to achieve a given accuracy.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.2 Energy estimation in VQE · p. 102
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.2 Energy estimation in VQE · p. 100
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.2 Energy estimation in VQE · p. 102
plaindirect- Asserted: passes — The authors put it forward as true.
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None of the sources cited for this claim have been checked yet.
In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.2 Energy estimation in VQE · p. 103
plaindirect- Asserted: passes — The authors put it forward as true.
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- Also stated in: § 5.1.2 Energy estimation in VQE · p. 103 · hedged
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.3 Optimization methods · p. 111
What the sources say
- Not checked yetcites [359] source
plainconditional- Asserted: passes — The authors put it forward as true.
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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
None of the sources cited for this claim have been checked yet.
In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.2 Energy estimation in VQE · p. 103
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.2 Energy estimation in VQE · p. 101
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.4 Physical realizations of VQE · p. 112
What the sources say
boosted- Asserted: passes — The authors put it forward as true.
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- Also stated in: § 5.1.4 Physical realizations of VQE · p. 112 · plain
In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > Strategies for reducing the number of measurements in Hamiltonian averaging. · p. 106
What the sources say
- Not checked yetcites [346] source
plaindirect- Asserted: passes — The authors put it forward as true.
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- Also stated in: § Strategies for reducing the number of measurements in Hamiltonian averaging. · p. 107 · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.2 Energy estimation in VQE · p. 101
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In the paper
“Other techniques that could be implemented using this strategy are canonical transformation theory 119 and perturbative explicit correlated corrections 363,364 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.6 Calculation of molecular properties with VQE · p. 116
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hedgedcorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“This matrix inversion and multiplication is carried out by a classical processor.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.2.1 Variational quantum simulator · p. 118
What the sources say
- Not checked yetcites [200] source
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.3.1 Adiabatic · p. 120
hedgeddirect- Asserted: passes — The authors put it forward as true.
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In the paper
“There are no guarantees on their performance, and their proper functioning can only be established on a case-by-case basis.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.2.3 Imaginary-time variational quantum simulator for excited states · p. 119
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.6 Calculation of molecular properties with VQE · p. 116
What the sources say
- Not checked yetcites [362] source
hedgedconditional- Asserted: passes — The authors put it forward as true.
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.6 Calculation of molecular properties with VQE · p. 116
What the sources say
- Not checked yetcites [346] source
hedgeddirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.3.1 Adiabatic · p. 120
What the sources say
- Not checked yetcites [60] Preskill, J. Quantum Computing in the NISQ Era and Beyond. 2018, arXiv:1801.00862 [quant-ph]; https://arxiv.org/abs/1801.00862. source
plain- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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:”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.6 Calculation of molecular properties with VQE · p. 116
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- Also stated in: § 5.1.6 Calculation of molecular properties with VQE · p. 116 · plain
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“An important caveat regarding these alternative algorithms is that, as with VQE, they are heuristic.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.2.3 Imaginary-time variational quantum simulator for excited states · p. 119
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
None of the sources cited for this claim have been checked yet.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.5 VQE for excited states · p. 114
What the sources say
- Not checked yetcites [320] source
- Not checked yetcites [58] Poulin D., 2014, Quantum Inf. Comput.
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- Also stated in: § 5.1.5 VQE for excited states · p. 114 · plain
None of the sources cited for this claim have been checked yet.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.5 VQE for excited states · p. 113
What the sources say
- Not checked yetcites [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.
- Not checked yetcites [316] source
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- Also stated in: § 5.1.5 VQE for excited states · p. 113 · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The results discussed so far are based on gate model quantum computers, which can be broadly categorized as the 'digital' approach to quantum simulation.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.3.1 Adiabatic · p. 120
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Thermal fluctuations which drive the system out of the ground state can be mitigated by keeping the system at extremely low temperatures.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.3.1 Adiabatic · p. 121
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper may give no reason why it holds.
In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.2.1 Variational quantum simulator · p. 118
What the sources say
- Not checked yetcites [200] source
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“A suitable adiabatic quantum computer would be able to effectively encode such a spin Hamiltonian.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.3.1 Adiabatic · p. 120
hedgedconditional- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper may give no reason why it holds.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.2.1 Variational quantum simulator · p. 117
What the sources say
- Not checked yetcites [200] source
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.5 VQE for excited states · p. 115
What the sources say
- Not checked yetcites [316] source
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
- Also stated in: § 5.1.5 VQE for excited states · p. 115 · plain
None of the sources cited for this claim have been checked yet.
In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.3.1 Adiabatic · p. 120
What the sources say
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.1.6 Calculation of molecular properties with VQE · p. 116
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“There are no guarantees on their performance, and their proper functioning can only be established on a case-by-case basis.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.2.3 Imaginary-time variational quantum simulator for excited states · p. 119
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.3.1 Adiabatic · p. 121
plain- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The use of quantum error correction also stands to improve the performance of variational quantum algorithms such as VQE.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 6 Summary and outlook · p. 130
hedgedconditional- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 6 Summary and outlook · p. 130
hedgedconditional- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“The quantum subspace expansion technique, developed in McClean et al. 320 , was motivated by the concept of linear response of the ground state.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 6 Summary and outlook · p. 129
What the sources say
- Not checked yetcites [320] source
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The goal is to compute the FCP by sampling from the output distribution of a modified boson sampling device, as shown in Figure 9.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.3.2 Linear optics · p. 122
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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).”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.3.2 Linear optics · p. 125
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
None of the sources cited for this claim have been checked yet.
In the paper
“In the past few years, several papers have proposed new quantum algorithms along with experimental demonstrations on quantum devices 387,388 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 6 Summary and outlook · p. 130
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.3.2 Linear optics · p. 122
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 6 Summary and outlook · p. 129
What the sources say
- Not checked yetcites [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
plaindirect- Asserted: passes — The authors put it forward as true.
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None of the sources cited for this claim have been checked yet.
In the paper
“Already, several methods have been introduced which aim to bridge the gap between these two classes of quantum algorithms 382-384 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 6 Summary and outlook · p. 130
What the sources say
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“In particular the proposal by Olivares et al. 379 consists of superconducting resonators interacting through tunable couplers.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.3.2 Linear optics · p. 125
What the sources say
- Not checked yetcites [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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In the paper
“With the arrival of usable quantum devices 385,386 , we have now entered a prototype era in the field of quantum algorithms.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 6 Summary and outlook · p. 130
What the sources say
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In the paper
“Superconducting circuits have no experimental realizations as of yet, however promising architectures have been theoretically proposed.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.3.2 Linear optics · p. 125
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.3.2 Linear optics · p. 125
What the sources say
- Not checked yetcites [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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In the paper
“But, more importantly, Hamiltonian simulation serves as a subroutine for other algorithms, such as the quantum phase estimation algorithm.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 6 Summary and outlook · p. 129
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.3.2 Linear optics · p. 124
What the sources say
- Not checked yetcites [376] source
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In the paper
“In the few years since its invention, there has been significant algorithmic development surrounding the variational quantum eigensolver algorithm 320,326,327,360 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 6 Summary and outlook · p. 128
What the sources say
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In the paper
“Although promising, this method requires the use of quantum error correction to properly function, and is therefore not feasible using near-term quantum devices.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 6 Summary and outlook · p. 128
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 6 Summary and outlook · p. 128
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.3.2 Linear optics · p. 122
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In the paper
“Nevertheless, other physical platforms such as trapped ions or superconducting circuits have proven useful in simulating molecular dynamics.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.3.2 Linear optics · p. 125
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In the paper
“This poses a challenge as molecular Hamiltonians and the adiabatic evolution require the spread of coupling term strengths over multiple orders of magnitude.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 5.3.1 Adiabatic · p. 121
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > C.3 Mapping the problem · p. 143
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In the paper
“| Superconducting processor (transmon qubits) | H 2 | 2 | 'Hardware- efficient' ansatz | SPSA | Ground state energy | 56 |”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > C.6 Appendix Glossary · p. 152
What the sources say
- Not checked yetcites [56] source
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > C.5 Variational quantum eigensolver for quantum chemistry · p. 149
What the sources say
- Not checked yetcites Section 5.1
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 6 Summary and outlook · p. 130
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > A Quantum chemistry basis sets · p. 137
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In the paper
“This is important for quantum simulation, as largerweight terms in the Hamiltonian require longer circuits for their simulation.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > B.2 Bravyi-Kitaev mapping · p. 139
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In the paper
“For example, in metals, valence electrons have a behavior more similar to free electrons.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > A Quantum chemistry basis sets · p. 136
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In the paper
“Choosing basis set functions with the right mathematical properties can significantly facilitate the evaluation of the Hamiltonian elements and expectation values.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > A Quantum chemistry basis sets · p. 132
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > A Quantum chemistry basis sets · p. 132
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In the paper
“For example, the variational quantum eigensolver algorithm can be viewed as a quantum version of Ritz's variational method.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 6 Summary and outlook · p. 131
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > C.5 Variational quantum eigensolver for quantum chemistry · p. 146
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In the paper
“GTOs have convenient analytical properties that allow for efficient schemes for the computational evaluation of molecular integrals.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > A Quantum chemistry basis sets · p. 135
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In the paper
“Similarly, tight functions (GTOs with large orbitals exponents) can be included when the description of core electrons plays an important role in the accuracy.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > A Quantum chemistry basis sets · p. 135
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > A Quantum chemistry basis sets · p. 132
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In the paper
“Atomic basis sets are well suited to describe electron densities in isolated molecules, where the densities resemble those of individual atoms.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > A Quantum chemistry basis sets · p. 136
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > B.2 Bravyi-Kitaev mapping · p. 140
What the sources say
- Not checked yetcites [402] source
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In the paper
“The quality of the basis set can be improved by decreasing the separation in the grid.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > A Quantum chemistry basis sets · p. 137
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > A Quantum chemistry basis sets · p. 134
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > A Quantum chemistry basis sets · p. 137
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > C.2 Defining the chemistry problem · p. 142
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In the paper
“Contracted GTOs have been adopted as the main basis set functions for electronic structure calculations in isolated molecules 399”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > A Quantum chemistry basis sets · p. 135
What the sources say
- Not checked yetcites [399] source
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In the paper
“Here, the parity operators are low-weight, while the occupation operators become high-weight.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > B.1 Jordan-Wigner mapping · p. 138
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > A Quantum chemistry basis sets · p. 134
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In the paper
“We note that this step is a general prerequisite for implementing Hamiltonian simulations, the quantum phase estimation algorithm, or the variational quantum eigensolver.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > C.3 Mapping the problem · p. 143
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In the paper
“PWs and their dual diagonalize the kinetic and potential operators, respectively.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > A Quantum chemistry basis sets · p. 137
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The existing basis sets in computational chemistry offer different levels of compromise among these desired qualities.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > A Quantum chemistry basis sets · p. 132
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > C.3 Mapping the problem · p. 143
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In the paper
“While the parity mapping is a valid transformation from fermionic operators to qubits”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > B.1 Jordan-Wigner mapping · p. 139
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In the paper
“However, the most optimistic estimates predict that quantum computers will solve useful problems in 2 to 5 years 395 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 6 Summary and outlook · p. 130
What the sources say
- Not checked yetcites [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.
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In the paper
“We note that for the H 2 example, this level of theory is equivalent with the exact solution.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > C.5 Variational quantum eigensolver for quantum chemistry · p. 147
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > A Quantum chemistry basis sets · p. 132
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In the paper
“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 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 6 Summary and outlook · p. 130
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In the paper
“While VQE is a near-term alternative to the quantum phase estimation algorithm due to its low coherence time requirements”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > C.5 Variational quantum eigensolver for quantum chemistry · p. 148
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“The electronic system, as written in the first-quantized picture in Equation 100, assumes an infinite-dimensional Hilbert space.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > C.2 Defining the chemistry problem · p. 142
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > A Quantum chemistry basis sets · p. 135
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“The more conservative estimates predict that this crossover will occur in 15 to 20 years 394 .”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 6 Summary and outlook · p. 130
What the sources say
- Not checked yetcites [394] source
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In the paper
“However, unlike STOs, they do not describe the cusp and exponential tails of the electronic density correctly.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > A Quantum chemistry basis sets · p. 135
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In the paper
“This points to the increasing value in having quantum algorithm developers working closely with quantum machines.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 6 Summary and outlook · p. 130
hedgeddirect- Asserted: passes — The authors put it forward as true.
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In the paper
“| | LiH | 4 | 'Hardware- efficient' ansatz | SPSA | Ground state energy | 56 |”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > C.6 Appendix Glossary · p. 152
What the sources say
- Not checked yetcites [56] source
plainSingle case- Asserted: passes — The authors put it forward as true.
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In the paper
“| 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 |”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > C.6 Appendix Glossary · p. 152
What the sources say
- Not checked yetcites [316] source
plainSingle case- Asserted: passes — The authors put it forward as true.
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In the paper
“| | BeH 2 | 6 | 'Hardware- efficient' ansatz | SPSA | Ground state energy | 56 |”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > C.6 Appendix Glossary · p. 152
What the sources say
- Not checked yetcites [56] source
plainSingle case- Asserted: passes — The authors put it forward as true.
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In the paper
“| Ion trap processor (Ca + ions) | H 2 | 2 | UCC | Grid scan and locally optimize | Ground state energy | 59 |”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > C.6 Appendix Glossary · p. 152
What the sources say
- Not checked yetcites [59] source
plainSingle case- Asserted: passes — The authors put it forward as true.
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In the paper
“| Superconducting processor (transmon qubits) | H 2 | 2 | Hardware- specific parametrized ansatz | PSO | Ground and excited state energies | 58 |”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > C.6 Appendix Glossary · p. 152
What the sources say
- Not checked yetcites [58] Poulin D., 2014, Quantum Inf. Comput.
plainSingle case- Asserted: passes — The authors put it forward as true.
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In the paper
“| | LiH | 3 | Approximate UCC | Grid scan and locally optimize | Ground state energy | 59 |”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > C.6 Appendix Glossary · p. 152
What the sources say
- Not checked yetcites [59] source
plainSingle case- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“| Superconducting processor (transmon qubits) via Cloud | Deuteron | 2-3 | UCC | Grid scan | Ground state energy | 330 |”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > C.6 Appendix Glossary · p. 152
What the sources say
- Not checked yetcites [330] source
plainSingle case- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
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?
The cited source says what the claim attributes to it.
In the citing paper
“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 cited source says what the claim attributes to it.
In the citing paper
“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
In the citing paper
“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]
In the citing paper
“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.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > B.1 Jordan-Wigner mapping · p. 138
“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”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > B.1 Jordan-Wigner mapping · p. 138
“where σ z = [ 1 0 0 -1 ] and σ + = [ 0 1 0 0 ] .”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.3 Going beyond classical limitations using quantum computers · p. 29
“In Equation 15, the notation σ i,j means the j -th operator in the i -th term of the Hamiltonian.”
§ Department of Physics, Harvard University, Cambridge, MA 02138, USA > 2.4 Chapter 2 Glossary · p. 34
“| σ i,j | i -th Pauli operator acting on the j -th qubit |”
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“of the LDCA circuit interleaves gates between even and odd neighboring qubits, U(k) MG = Y i2odd G(k) i;i+1 Y i2even G(k) i;i+1: (59) There are N 2 cycles in total, i.e. UNN MG = dM Y2 e k=1 U(k) MG; (60) such that the transformation Wcan be composed as W= UNN MG YM i=1 RZ i: (61) Dallaire-Demers et al.326 suggest tha…”
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Metadata
- 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
- 368
- 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
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