Quantum computational chemistry

The cost of storing the wavefunction scales exponentially with the number of electrons, N.

“However, the cost of storing the wavefunction scales exponentially with the number of electrons, N .”

The diagonalised 1-RDM for H2 in the cc-PVDZ basis is Diag(1.96588, 0.00611, 0.02104, 0.0002, 0.00001, 0.00314, 0.00314, 0.00016, 0.00016, 0.00016).

“The diagonalised 1-RDM is given by Diag (1.96588, 0.00611, 0.02104, 0.0002, 0.00001, 0.00314, 0.00314, 0.00016, 0.00016, 0.00016).”

The 5th orbital in the cc-PVDZ H2 1-RDM has a NOON that is 20 times smaller than the next smallest NOON.

“We can see that the 5 th orbital has a NOON that is 20 times smaller than the next smallest NOON.”

The differences in energy between the three minima of the H2 ground state dissociation curves in STO-3G, 6-31G and cc-PVDZ bases are considerably larger than chemical accuracy (1.6 mHartree).

“We can see that the differences in energy between the three minima are considerably larger than chemical accuracy (1 . 6 mHartree).”

Applying the simplified UCCSD operator to the HF state can reproduce the ground state given by Eq. (98).

“which can reproduce the ground state given by Eq. (98).”

The exact application of quantum mechanics to chemistry leads to equations too complicated to be soluble.

“The exact application of these laws leads to equations much too complicated to be soluble.”

• cites Dirac, 1929Not checked yet

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With only single qubit operations and CNOT gates, it is possible to approximate an arbitrary multi-qubit gate to any desired accuracy.

“With only single qubit operations and CNOT gates, it is possible to approximate an arbitrary multi-qubit gate to any desired accuracy (DiVincenzo, 1995).”

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Classical computers are unable to exactly simulate quantum systems in an efficient way.

“This leaves classical computers unable to exactly simulate quantum systems in an efficient way.”

Quantum computers are able to efficiently simulate quantum systems.

“As quantum computers are able to efficiently simulate quantum systems”

• cites Feynman, 1982Not checked yet

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The complexity of the wavefunction of a quantum system grows exponentially with the number of particles.

“The problem described by Dirac is that the complexity of the wavefunction of a quantum system grows exponentially with the number of particles.”

• cites Dirac, 1929Not checked yet

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Classical computational methods' ability to guide design is often precluded by the computational complexity of realistic models.

“However, their ability to guide design is often precluded by the computational complexity of realistic models.”

Physical error rates below the surface code threshold of around 1% have been achieved for trapped ion and superconducting qubits.

“Physical error rates below the surface code threshold of around 1 % (Fowler et al. , 2012b; Stephens, 2014; Wang et al. , 2011) have been achieved for trapped ion (Ballance et al. , 2016; Gaebler et al. , 2016) and superconducting (Arute et al. , 2019; Barends et al. , 2014) qubits.”

• cites Fowler et al., 2012bNot checked yet

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• cites Ballance et al., 2016Not checked yet

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• cites Arute et al., 2019Not checked yet

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Solving the ground state problem for a completely general local Hamiltonian is QMA-complete.

“Moreover, it is important to note that solving the ground state problem for a completely general local Hamiltonian is QMA-complete (quantum Merlin-Arthur complete, the quantum analogue of NP-complete) (Cubitt and Montanaro, 2016; Kempe et al. , 2006).”

• cites Cubitt and Montanaro, 2016Not checked yet

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• cites Kempe et al., 2006Not checked yet

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This makes it classically intractable to simulate more than a few particles on a grid using a classical computer.

“This makes it classically intractable to simulate more than a few particles on a grid using a classical computer.”

If the energy is known to chemical accuracy, then the chemical reaction rate at room temperature can be predicted to within an order of magnitude using the Eyring equation.

“If the energy is known to chemical accuracy, then the chemical reaction rate at room temperature can be predicted to within an order of magnitude using the Eyring equation (Evans and Polanyi, 1935; Eyring, 1935)”

• cites Evans and Polanyi, 1935Not checked yet

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• cites Eyring, 1935Not checked yet

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The reaction mechanism of the nitrogenase enzyme is not yet fully understood.

“While there has been significant experimental progress in understanding the structure of the nitrogenase enzyme over the last 100 years, the reaction mechanism is not yet fully understood (Burgess and Lowe, 1996).”

• cites Burgess and Lowe, 1996Not checked yet

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The ground state problem can also be tackled using variational algorithms.

“The ground state problem can also be tackled using variational algorithms (Peruzzo et al. , 2014), which we will discuss in detail in Sec. V.B.”

• cites Peruzzo et al., 2014Not checked yet

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Time evolution of open and closed quantum systems can be simulated using variational approaches.

“It has also been shown that time evolution of open and closed quantum systems can be simulated using variational approaches (Chen et al. , 2019; Endo et al. , 2018b; Heya et al. , 2019; Li and Benjamin, 2017).”

• cites Chen et al., 2019Not checked yet

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• cites Endo et al., 2018bNot checked yet

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• cites Li and Benjamin, 2017Not checked yet

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With physical error rates below the surface code threshold, around 10^3-10^4 physical qubits per logical qubit would be required to perform interesting tasks in a fault-tolerant manner.

“However, with these error rates, we would require around 10 3 -10 4 physical qubits per logical qubit to perform interesting tasks in a fault-tolerant manner (Campbell et al. , 2017; Fowler et al. , 2012a; O'Gorman and Campbell, 2017).”

• cites Campbell et al., 2017Not checked yet

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• cites Fowler et al., 2012aNot checked yet

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• cites O'Gorman and Campbell, 2017Not checked yet

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The property of convergence to the correct energy from above as the number of basis functions tends to infinity does not hold for grid based methods.

“This property does not hold for grid based methods.”

The problems solved by quantum computers with more than around 50 qubits are typically artificially constructed examples, rather than real-world problems.

“However, these problems are typically artificially constructed examples, rather than real-world problems (Boixo et al. , 2018; Harrow and Montanaro, 2017).”

• cites Boixo et al., 2018Not checked yet

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• cites Harrow and Montanaro, 2017Not checked yet

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A complete understanding of even the Fermi-Hubbard model is still elusive.

“While more complex models are likely required to fully understand the mechanism of cuprate superconductivity, a complete understanding of even the Fermi-Hubbard model is still elusive.”

Problems in the QMA-complete complexity class are not believed to be efficiently solvable with either a classical or quantum computer.

“Problems in this complexity class are not believed to be efficiently solvable with either a classical or quantum computer.”

A complete analytic or computational understanding of the copper-oxygen layers in cuprate superconductors is beyond the capabilities of current classical techniques.

“A complete analytic or computational understanding of these layers is beyond the capabilities of current classical techniques.”

The number of terms in the Hamiltonian scales polynomially with the number of particles for systems of interest in chemistry, due to the two-body nature of the Coulomb interactions between particles.

“The number of terms in the Hamiltonian scales polynomially with the number of particles for systems of interest in chemistry, due to the two-body nature of the Coulomb interactions between particles (Helgaker et al. , 2014).”

• cites Helgaker et al., 2014Not checked yet

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Many transition metal catalysts show strong correlation, often due to their open-shell nature and spatially degenerate states.

“However, many of these systems show strong correlation, often due to their openshell nature, and spatially degenerate states (Lyakh et al. , 2012).”

• cites Lyakh et al., 2012Not checked yet

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Finding the low lying energy levels of a quantum Hamiltonian is in general an exponentially difficult problem for classical computers.

“Finding the low lying energy levels of a quantum Hamiltonian is in general an exponentially difficult problem for classical computers.”

This strong correlation often precludes the study of transition metal catalysts in silico, forcing expensive, trial-and-error based discovery.

“This strong correlation often precludes their study in silico , forcing us to carry out expensive, trial-and-error based discovery -especially for transition metal based biological catalysts (Podewitz et al. , 2011).”

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Models of fermions hopping on a 2D square lattice, such as the one-band Fermi-Hubbard model and the related t-J model, appear to reproduce many experimental observations.

“In particular, models of fermions hopping on a 2D square lattice, such as the one-band Fermi-Hubbard model (Hubbard, 1963), and the related t-J model, appear to reproduce many experimental observations (Anderson, 2002; Lee et al. , 2006).”

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• cites Lee et al., 2006Not checked yet

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Mapping out potential energy curves explicitly is exponentially costly in the degrees of freedom of the molecule.

“Mapping out these potential energy curves explicitly is exponentially costly in the degrees of freedom of the molecule, and that there are a variety of methods being developed to solve this difficult problem more efficiently (Christiansen, 2012).”

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To the best of the authors' knowledge, only the work by Berry et al. (2019a) considers a chemical reaction dynamics calculation on a quantum computer using basis set methods.

“To the best of our knowledge, only the work by Berry et al. (2019a) considers a chemical reaction dynamics calculation on a quantum computer using basis set methods.”

• cites Berry et al., 2019aNot checked yet

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Mapping out potential energy surfaces is exponentially costly.

“As mentioned in the previous section, mapping out these potential energy surfaces is exponentially costly.”

Basis set methods exploit knowledge of the general spatial form of the orbitals to reduce the resources needed to simulate chemical systems.

“In contrast, basis set methods exploit this knowledge to reduce the resources needed to simulate chemical systems.”

If the electron-electron Coulomb interaction term is neglected, we obtain a new Hamiltonian which describes the behaviour of N independent electrons.

“If the electronelectron Coulomb interaction term in Eq. (17) is neglected, we obtain a new Hamiltonian which describes the behaviour of N independent electrons.”

A mean-field approximation (the Hartree-Fock procedure) can be used to obtain more suitable molecular orbitals.

“Instead, a mean-field approximation (the HartreeFock procedure, to be described in Sec. III.C.1) can be used to obtain more suitable molecular orbitals.”

MCSCF and CASSCF calculations are among the most effective classical methods at treating systems with strong static correlation.

“MCSCF and CASSCF calculations are among the most effective classical methods at treating systems with strong static correlation (Szalay et al. , 2012).”

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The eigenstates of the true Hamiltonian can be expressed as linear combinations of Slater determinants, with complex coefficients called determinant amplitudes.

“Consequently, the eigenstates of the true Hamiltonian can be expressed as linear combinations of these Slater determinants, written as”

• cites Helgaker et al. , 2014Not checked yet

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• cites Szabo and Ostlund, 2012Not checked yet

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As first quantised operators conserve the number of electrons, the second quantised operators must contain an equal number of creation and annihilation operators.

“As first quantised operators conserve the number of electrons, the second quantised operators must contain an equal number of creation and annihilation operators.”

Truncated CI calculations fail to satisfy size consistency.

“Truncated CI calculations also fail to satisfy a related property, known as 'size consistency'.”

Slater determinants, as eigenstates of a Hermitian operator, form a complete basis of the problem Hilbert space.

“As they are eigenstates of a Hermitian operator, these Slater determinants form a complete basis of the problem Hilbert space.”

MCSCF can be considered the best approximation to the exact wavefunction for a given number of determinants.

“MCSCF can be considered the best approximation to the exact wavefunction for a given number of determinants (Wang et al. , 2008).”

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Wavefunctions displaying dynamic correlation are often dominated by the Hartree-Fock determinant and have small additional contributions from a potentially large number of excited state determinants.

“Wavefunctions displaying dynamic correlation are often dominated by the Hartree-Fock determinant, and have small additional contributions from a (potentially large) number of excited state determinants.”

Static correlation often dominates in many systems of scientific interest, such as excited states or transition metals.

“As a result, static correlation often dominates in many systems of scientific interest, such as excited states (Lischka et al. , 2018), or transition metals (Vogiatzis et al. , 2019).”

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• cites Vogiatzis et al. , 2019Not checked yet

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It is not possible to perform a complete MCSCF calculation on all possible determinants for systems with more than a few electrons, as the number of determinants scales exponentially with the number of electrons.

“It is not possible to perform a complete MCSCF calculation on all possible determinants for systems with more than a few electrons, as the number of determinants scales exponentially with the number of electrons.”

Because low energy excitations dominate the ground state wavefunction in many physical systems, truncated CI can still produce good approximations to the ground state energy.

“However, as low energy excitations dominate the ground state wavefunction in many physical systems, these truncations can still produce good approximations to the ground state energy (Helgaker et al. , 2014; Szabo and Ostlund, 2012).”

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• cites Szabo and Ostlund, 2012Not checked yet

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In the case of static correlation, the Hartree-Fock method provides a poor approximation to the ground state wavefunction.

“In this case, the Hartree-Fock method provides a poor approximation to the ground state wavefunction.”

It is in general not possible to avoid strong static correlation when considering the whole of the potential energy surface.

“Lyakh et al. (2012) note that, while it is sometimes possible to reduce the level of static correlation by manually enforcing correct spin symmetries, and using appropriately localised orbitals, it is in general not possible to avoid strong static correlation when considering the whole of the potential energy surface.”

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The expectation values of second quantised operators must be equivalent to the expectation values of the corresponding first quantised operators.

“Therefore, the expectation values of second quantised operators must be equivalent to the expectation values of the corresponding first quantised operators.”

The linear-combination-of-Slater-determinants solutions are exact provided that the molecular orbitals form a complete basis for the single-particle states and the N-electron wavefunction contains all determinants that these molecular orbitals can generate.

“These solutions are exact, provided that the molecular orbitals form a complete basis for the single particle states, and the N -electron wavefunction contains all of the determinants that these molecular orbitals can generate (Helgaker et al. , 2014; Szabo and Ostlund, 2012).”

• cites Helgaker et al. , 2014Not checked yet

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• cites Szabo and Ostlund, 2012Not checked yet

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The CI method converges slowly to the full configuration interaction wavefunction as a result of its linear parametrization.

“The method converges slowly to the full configuration interaction wavefunction, as a result of its linear parametrization.”

The energy obtained from a truncated CI calculation is not size extensive.

“The energy obtained from a truncated CI calculation is not size extensive.”

• cites Lyakh et al. , 2012Not checked yet

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As a result, the Hamiltonian contains up to M^4 terms, depending on the basis functions used.

“As a result, the Hamiltonian contains up to M 4 terms, depending on the basis functions used.”

The presence of strong static correlation can be evidenced by multiple near-degenerate solutions of the Hartree-Fock procedure.

“The presence of strong static correlation can be evidenced by multiple near-degenerate solutions of the Hartree-Fock procedure.”

The four classical methods reviewed for approximating the ground state wavefunction with a restricted number of Slater determinants create parametrized trial states that can be optimised to approach the ground state to an accuracy determined by the approximations made.

“These methods create parametrized trial states, which can then be optimised to approach the ground state (to an accuracy determined by the approximations made).”

The Hamiltonian only contains terms with up to four creation and annihilation operators because the Coulomb interaction between the electrons is a two-body interaction.

“The Hamiltonian only contains terms with up to four creation and annihilation operators (two creation, two annihilation) because the Coulomb interaction between the electrons is a two-body interaction.”

The plane wave and plane wave dual basis sets diagonalise the kinetic and potential operators, respectively.

“These basis sets diagonalise the kinetic and potential operators, respectively.”

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In the Jordan-Wigner encoding, a fermionic operator mapped to qubits generally has a weight of O(M) Pauli operators, each acting on a different qubit.

“The string of Z operators means that a fermionic operator mapped to qubits generally has a weight of O ( M ) Pauli operators, each acting on a different qubit.”

To describe molecular systems, approximately 10 to 100 times as many plane wave basis functions are required as GTOs.

“However, to describe molecular systems, approximately 10 to 100 times as many plane wave basis functions are required as GTOs (Babbush et al. , 2018c).”

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Certain orbitals are sometimes very likely to be either occupied or virtual in all Slater determinants in the wavefunction.

“It is sometimes the case that certain orbitals are very likely to be either occupied or virtual in all Slater determinants in the wavefunction.”

Spin-orbitals can be removed from the calculation if their expected occupation number is close to 0 or 1, reducing the calculation to only the most important (ambiguously occupied) orbitals, known as a reduced active space.

“Specifically, we are able to remove spin-orbitals from the calculation if their expected occupation number is close to 0 or 1. Our calculation is reduced to including only the most important (ambiguously occupied) orbitals. This is known as performing the calculation in a reduced active space.”

The wavefunction generated by the canonical CC method does not obey the Rayleigh-Ritz variational principle.

“Most notably, the wavefunction generated by the canonical CC method does not obey the Rayleigh-Ritz variational principle (Helgaker et al. , 2014).”

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Takeshita et al. (2019) showed how to re-integrate some of the energy contribution of virtual orbitals into a quantum simulation without requiring additional qubits to represent the virtual orbitals, using increased measurements and classical post-processing.

“Takeshita et al. (2019) showed how to re-integrate some of the energy contribution of the virtual orbitals into a quantum simulation, without requiring additional qubits to represent the virtual orbitals. Their method utilises an increased number of measurements, and classical post-processing.”

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cc-PVnZ basis sets with higher n contain orbitals that better approximate the true atomic orbitals than those with lower n values.

“cc-PV n Z basis sets with higher values of n contain orbitals that better approximate the true atomic orbitals than those with lower n values.”

Conventional orbital basis sets are not necessarily optimal for calculations performed on quantum computers.

“While the aforementioned basis sets have a long history of use in classical computational chemistry (and as a result, early work in quantum computational chemistry), they are not necessarily optimal basis sets for calculations performed on quantum computers.”

The limitation of finite basis sets can be overcome by measuring the ground state energy in several different cc-PVnZ bases and extrapolating to the basis set limit.

“This limitation can be overcome by measuring the ground state energy in several different cc-PV n Z bases, and then extrapolating to the basis set limit.”

Using a quantum computer, the full configuration interaction wavefunction can be stored using only M qubits.

“However, using a quantum computer, we can instead store the full configuration interaction (FCI) wavefunction using only M qubits (Aspuru-Guzik et al. , 2005).”

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The BK-tree mapping produces qubit operators with a greater weight than the standard BK mapping.

“It has been noted that the BK-tree mapping produces qubit operators with a greater weight than the standard BK mapping (Sung, 2018).”

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This reduction in Hamiltonian terms leads to a reduction in the asymptotic scaling of quantum chemistry algorithms to find the ground state energy of molecules and solid state systems.

“This in turn leads to a reduction in the asymptotic scaling of quantum chemistry algorithms to find the ground state energy of molecules and solid state systems”

Locality preserving mappings typically require more qubits than the JW encoding.

“These mappings typically require more qubits than the JW encoding.”

Grid based approaches are typically considered unsuitable for near-term quantum computers, which will have relatively few qubits.

“Consequently, grid based approaches are typically considered unsuitable for near-term quantum computers, which will have relatively few qubits.”

When carrying out a HF calculation in an STOnG basis, the true HF energy will not be obtained, as the STOnG basis sets only approximate the true HF orbitals.

“It is important to note that when carrying out a HF calculation in an STOn G basis, the true HF energy (i.e. the energy obtained by performing a HF calculation using a grid based method, on an infinitely precise grid) will not be obtained, as the STOn G basis sets only approximate the true HF orbitals.”

Applying the Bravyi-Kitaev mapping to a fermionic operator results in a qubit operator with a Pauli weight of O(log2 M).

“Applying the BK mapping to a fermionic operator results in a qubit operator with a Pauli weight of O (log 2 M ).”

Even large (n=5) basis sets struggle to exactly represent the true HF orbitals of simple molecules such as N2.

“However, even large ( n = 5) basis sets struggle to exactly represent the true HF orbitals of simple molecules such as N 2 (Helgaker et al. , 2014).”

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The number of terms in the Hamiltonian plays a key role in the cost of some quantum chemistry algorithms.

“As we will see in Sec. V, the number of terms in the Hamiltonian plays a key role in the cost of some quantum chemistry algorithms.”

Phase estimation can be used to find the lowest energy eigenstate and excited states of a physical Hamiltonian.

“Phase estimation (Kitaev, 1995) can be used to find the lowest energy eigenstate, | E 0 〉 , and excited states, | E i> 0 〉 , of a physical Hamiltonian (Abrams and Lloyd, 1999).”

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• cites Abrams and Lloyd, 1999Not checked yet

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Phase estimation can become exponentially costly due to imperfect preparation of eigenstates of non-interacting subsystems.

“Even more problematically, McClean et al. (2014) showed that phase estimation can become exponentially costly, by considering the imperfect preparation of eigenstates of non-interacting subsystems.”

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The accuracy of product formula simulations under a time-dependent Hamiltonian depends on the derivatives of the Hamiltonian.

“Wiebe et al. (2011) showed that the accuracy of such simulations depends on the derivatives of the Hamiltonian (although this dependence may be alleviated by incorporating randomisation procedures (Poulin et al. , 2011)).”

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• cites Poulin et al., 2011Not checked yet

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The electronic structure Hamiltonian in the canonical molecular orbital basis contains O(M^4) terms, where M is the number of spin-orbitals.

“As discussed in Sec. III.A, the electronic structure Hamiltonian in the canonical molecular orbital basis contains O ( M 4 ) terms, where M is the number of spinorbitals in the molecule.”

This has caused second quantised basis set methods to become the de facto option for experimental demonstrations of quantum computational chemistry algorithms, due to the limits of current quantum hardware.

“This has caused second quantised basis set methods to become the de facto option for experimental demonstrations of quantum computational chemistry algorithms, due to the limits of current quantum hardware.”

Qubitization naturally implements a quantum walk operator with eigenvalues e^{-i arcsin(E_k/α)} where E_k is the kth eigenvalue of the Hamiltonian H and α is a normalisation factor.

“Qubitization naturally implements a quantum walk operator with eigenvalues e -i arcsin( E k /α ) where E k is the k th eigenvalue of the Hamiltonian H and α is a normalisation factor.”

It is necessary to develop state preparation routines that result in at worst a polynomially decreasing overlap with the FCI ground state as system size increases.

“This highlights the necessity of developing state preparation routines which result in at worst a polynomially decreasing overlap with the FCI ground state, as the system size increases.”

In the Bravyi-Kitaev superfast transform, the number of gates required to implement a fermionic operator scales as O(d) where d is the degree of the graph.

“However, the number of gates required to implement a fermionic operator will scale as O ( d ) where d is the degree of the graph.”

For any Hamiltonian H_s, a state close to its ground state can be prepared via adiabatic state preparation.

“For any Hamiltonian H s , we can prepare a state | Ψ 〉 that is close to its ground state via adiabatic state preparation (Albash and Lidar, 2018).”

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Gaussian orbitals result in a second quantised Hamiltonian with O(M^4) terms.

“They result in a second quantised Hamiltonian with O ( M 4 ) terms.”

Huggins et al. (2019b) used low rank decomposition to reduce the number of circuit repetitions required for variational algorithms to find the ground state energy of chemical systems.

“This low rank decomposition technique was also used by Huggins et al. (2019b), to reduce the number of circuit repetitions required for variational algorithms to find the ground state energy of chemical systems.”

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If an initial state with sufficiently large overlap with the ground state is available, adiabatic state preparation may be forgone and phase estimation carried out directly on that initial state.

“We note however that if an initial state with sufficiently large overlap with the ground state is available, we may be able to forgo adiabatic state preparation entirely, and instead carry out phase estimation directly on that initial state.”

It is difficult to optimally allocate resources between the error budgets ε_PE, ε_U, and ε_CS in order to minimise the total error.

“In general, it is difficult to optimally allocate resources between these error budgets in order to minimise the total error (Kivlichan et al. , 2019a; Reiher et al. , 2017).”

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• cites Reiher et al., 2017Not checked yet

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The ultimate limit for the number of calls required to e^{-iH} is π/ε_PE for a completely general Hamiltonian H, approximately obtained using Bayesian approaches or entanglement based approaches.

“The ultimate limit for the number of calls required to e -iH is π/ϵ PE (for a completely general Hamiltonian, H ), which is approximately obtained using Bayesian approaches (Berry et al. , 2009; Higgins et al. , 2007; Paesani et al. , 2017; Wiebe and Granade, 2016), or entanglement based approaches (Babbush et al. , 2018b).”

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• cites Higgins et al. , 2007Not checked yet

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• cites Babbush et al. , 2018bNot checked yet

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To achieve accuracy ε with first-order Trotterization, the number of Trotter steps S = O(t^2/ε) should be large.

“In practice, to achieve accuracy ε , the number of Trotter steps S = O ( t 2 /ε ) should be large in order to suppress the errors in the approximation.”

The finite precision ε_PE obtained in E0 is not the only source of error in phase estimation.

“The finite precision, ϵ PE , obtained in E 0 is not the only source of error in the algorithm.”

For a Trotter decomposition of e^{-iH}, the error in the energy eigenvalue obtained from phase estimation is upper bounded by ε_PE + ε_U + ε_CS.

“For the specific case of a Trotter decomposition of e -iH , Reiher et al. (2017) showed that the error in the energy eigenvalue obtained from phase estimation is upper bounded by ϵ PE + ϵ U + ϵ CS .”

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In order to account for c0<1, the number of repetitions of phase estimation must be multiplied by 1/|c0|^2, on average, to obtain the ground state.

“In order to account for the fact that c 0 &lt; 1, we must multiply the number of repetitions of phase estimation by 1 / | c 0 | 2 , on average, to obtain the ground state.”

Randomisation procedures improve the accuracy obtained using product formulae.

“Randomisation procedures (such as randomly ordering the terms in the Trotter sequence, or stochastically choosing which terms to include in the Hamiltonian), have been shown to improve the accuracy obtained using product formulae (Campbell, 2018; Childs et al. , 2018; Ouyang et al. , 2019).”

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• cites Childs et al., 2018Not checked yet

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A series of improvements reduced the scaling of phase estimation for molecules from O(M^11) to (empirically) O(M^5).

“A series of improvements throughout these papers reduced the scaling of phase estimation for molecules from O ( M 11 ) (Wecker et al. , 2014) to (empirically) O ( M 5 ) (Babbush et al. , 2015).”

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Higher order product formulae scale better with respect to simulation error than the first order method.

“It is also possible to use higher order product formulae (Berry et al. , 2007; D¨ ur et al. , 2008; Suzuki, 1976), which scale better with respect to the simulation error than the first order method.”

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Initialising the qubit register in a state with sufficiently large overlap with the target eigenstate is a non-trivial problem.

“Initialising the qubit register in a state which has a sufficiently large overlap with the target eigenstate (typically the ground state), is a non-trivial problem.”

The method by Bravyi et al. (2017) utilises conservation of electron number and spin to enable systematic reduction of two qubits when using the parity, BK (with the caveat that the number of orbitals is a power of two), or BK-tree encoding.

“Here, we will describe the method by Bravyi et al. (2017), which utilises two such symmetries: conservation of electron number and spin. This method enables the systematic reduction of two qubits when using the parity, BK (with the caveat that the number of orbitals is a power of two), or BK-tree encoding.”

• cites Bravyi et al. (2017)Not checked yet

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Annealing times for methylene (CH2) could be reduced by up to four orders of magnitude by using an initial state with larger overlap with the true ground state.

“This view is supported by the numerical simulations of Veis and Pittner (2014), who found that annealing times for methylene (CH 2 ) could be reduced by up to four orders of magnitude by using an initial state with larger overlap with the true ground state.”

• cites Veis and Pittner, 2014Not checked yet

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Introducing additional 'driving' Hamiltonians is a possible method for reducing the annealing time required.

“One possible method for reducing the annealing time required is to introduce additional 'driving' Hamiltonians, as was numerically investigated by Matsuura et al. (2018).”

• cites Matsuura et al., 2018Not checked yet

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The efficiency of adiabatic state preparation depends on the gap between the ground state and the first excited state along the path between H_0 and H_s.

“The efficiency of adiabatic state preparation depends on the gap between the ground state and the first excited state along the path between H 0 and H s .”

Errors arise from imperfect implementation of the controlled unitary evolutions applied to the main register (ε_U).

“There are also errors arising from imperfect implementation of the controlled unitary evolutions applied to the main register, which we denote as ϵ U .”

Rigorous bounds on the Trotter error are believed to be loose by several orders of magnitude.

“A notable feature of these Trotter-based simulations is that rigorous bounds on the Trotter error are believed to be loose by several orders of magnitude, which may impact the scaling of these approaches.”

• cites Babbush et al., 2015Not checked yet

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• cites Poulin et al., 2015Not checked yet

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A randomly chosen state would have an exponentially vanishing probability of collapsing to the desired ground state as system size increases.

“This is important, because a randomly chosen state would have an exponentially vanishing probability of collapsing to the desired ground state, as the system size increases.”

The Trotter error bounds appear to be loose by several orders of magnitude.

“While the advanced Hamiltonian simulation methods described above are asymptotically more efficient than Trotterization, the Trotter error bounds appear to be loose by several orders of magnitude (Babbush et al. , 2015; Poulin et al. , 2015).”

• cites Babbush et al., 2015Not checked yet

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• cites Poulin et al., 2015Not checked yet

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The VQE has been experimentally demonstrated on superconducting qubits, trapped ions, and photonic systems.

“The VQE has been experimentally demonstrated on most leading quantum architectures including: superconducting qubits (Colless et al. , 2018; Kandala et al. , 2017; O'Malley et al. , 2016), trapped ions (Hempel et al. , 2018; Kokail et al. , 2019; Nam et al. , 2019; Shen et al. , 2017), and photonic systems (Peruzzo et al. , 2014), and shows many desirable features.”

• cites Colless et al., 2018Not checked yet

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• cites Kandala et al., 2017Not checked yet

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• cites O'Malley et al., 2016Not checked yet

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Babbush et al. (2018a) obtained a scaling of O(N^{4/3} M^{2/3} + N^{8/3} M^{1/3}) for first quantized plane wave basis.

“Babbush et al. (2018a) obtained a scaling of O ( N 4 / 3 M 2 / 3 + N 8 / 3 M 1 / 3 ), similar to the scaling of their interaction picture algorithm discussed above.”

• cites Babbush et al., 2018aNot checked yet

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Alternative methods are required for near-future chemistry simulation.

“Consequently, alternative methods are required for near-future chemistry simulation.”

The scaling of the CI matrix based algorithm of Babbush et al. (2017) was reduced from O(N^2 M^3) in a Gaussian basis to O(M^{2.67}) in a plane wave basis.

“The scaling of the CI matrix based algorithm of Babbush et al. (2017) was reduced from O ( N 2 M 3 ) in a Gaussian basis to O ( M 2 . 67 ) in a plane wave basis (Babbush et al. , 2018c).”

• cites Babbush et al., 2017Not checked yet

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• cites Babbush et al., 2018cNot checked yet

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The canonical UCCSD implementation requires O(M^3 N^2) gates when using the Jordan-Wigner mapping, where M is the number of spin-orbitals and N is the number of electrons.

“The canonical UCCSD implementation requires O ( M 3 N 2 ) gates when using the Jordan-Wigner mapping (where M is the number of spin-orbitals, and N is the number of electrons) (Romero et al. , 2019).”

• cites Romero et al., 2019Not checked yet

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• AssumptionThe authors state this as an assumption: A single Trotter step can be used to implement the UCC operator.

A study of spin Hamiltonians found that the asymptotic scaling of Trotter methods was much worse than qubitization + quantum signal processing and Taylor series methods.

“A study of spin Hamiltonians (Childs et al. , 2017) found that the asymptotic scaling of Trotter methods was much worse than the qubitization + quantum signal processing, and Taylor series methods.”

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Kivlichan et al. (2017) obtained a scaling of O((N/h^2 + N^2)t) for grid based methods.

“Those authors obtained a scaling of O (( N/h 2 + N 2 ) t ) (excluding logarithmic factors), where h is the grid spacing and t is the simulation time.”

• cites Kivlichan et al., 2017Not checked yet

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For certain functions f(H), qubitization can be used on its own to directly implement the unitary e^{-if(H)t} without approximation errors.

“However, it was later noted that for certain functions f ( H ), qubitization could be used on its own to directly implement the unitary e -if ( H ) t without approximation errors (Berry et al. , 2018; Poulin et al. , 2018).”

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• cites Poulin et al., 2018Not checked yet

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For 3D materials, the scaling of Trotter based approaches to phase estimation in a plane wave dual basis was reduced to around O(M^2).

“For 3D materials, such as diamond, the scaling of Trotter based approaches to phase estimation in a plane wave dual basis was reduced to around O ( M 2 ).”

Babbush et al. (2018b) obtained a gate scaling of O(M) for the Fermi-Hubbard model using qubitization.

“Those authors obtained a gate scaling of O ( M ) for the Fermi-Hubbard model.”

• cites Babbush et al., 2018bNot checked yet

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Wecker et al. (2015b) found that their approach to finding the ground state of the Fermi-Hubbard model required O(M^3) gates.

“Their approach to finding the ground state required O ( M 3 ) gates.”

• cites Wecker et al., 2015bNot checked yet

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When considering an intensive error in the energy, the qubitization approach outperforms the Trotter-based method of Kivlichan et al. (2019a).

“When considering an intensive error in the energy, this approach outperforms the Trotter-based method of Kivlichan et al. (2019a).”

• cites Kivlichan et al., 2019aNot checked yet

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The Hamiltonian has a sparsity of O(N^2 M^2) in the CI matrix basis.

“The Hamiltonian has a sparsity of O ( N 2 M 2 ) in this basis.”

When numerical simulations were performed, Trotter methods required lower gate counts than the other methods.

“However, when numerical simulations were performed, Trotter methods required lower gate counts than the other methods.”

Kivlichan et al. (2019a) found that the cost of Fermi-Hubbard model simulations was effectively between O(1) and O(M^{1/2}) using Trotterisation when targeting extensive error.

“This was subsequently improved by Kivlichan et al. (2019a), who considered the case where an extensive error in the energy is targeted, and found that the cost of Fermi-Hubbard model simulations was effectively between O (1) and O ( M 1 / 2 ) using Trotterisation.”

• cites Kivlichan et al., 2019aNot checked yet

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The VQE is a promising algorithm for near-term quantum hardware.

“A promising algorithm for near-term quantum hardware is the variational quantum eigensolver (VQE), first proposed and experimentally realised by Peruzzo et al. (2014), and elaborated on by McClean et al. (2016).”

• cites Peruzzo et al., 2014Not checked yet

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• cites McClean et al., 2016Not checked yet

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The best product based formulae approaches scale as O(M^{1.67} N^{1.83}) in a plane wave basis.

“The best product based formulae approaches described above scale as O ( M 1 . 67 N 1 . 83 ) in a plane wave basis (Babbush et al. , 2018c).”

• cites Babbush et al., 2018cNot checked yet

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The VQE exchanges the long coherence times needed for phase estimation for a polynomial overhead due to measurement repetitions and classical processing.

“This exchanges the long coherence times needed for phase estimation for a polynomial overhead due to measurement repetitions and classical processing.”

Childs and Su (2019) proved that a kth order product formula can simulate time evolution of an M qubit lattice Hamiltonian using O((Mt)^{1+1/k}) elementary operations.

“Similar results were obtained by Childs and Su (2019), who proved that a k th order product formula can simulate time evolution of an M qubit lattice Hamiltonian using O (( Mt ) 1+ 1 k ) elementary operations.”

• cites Childs and Su, 2019Not checked yet

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The Hamiltonian averaging method requires O(1/ε^2) measurements to determine the energy to a precision ε.

“This is known as the Hamiltonian averaging method of calculating the energy (McClean et al. , 2014), and requires O (1 /ϵ 2 ) measurements to determine the energy to a precision ϵ .”

• cites McClean et al., 2014Not checked yet

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This effect becomes exponentially more prominent as the number of qubits and circuit depth increases.

“This effect becomes exponentially more prominent as the number of qubits and circuit depth increases.”

Berry et al. (2019b) produced an algorithm with scaling O(M^{1.5} λ), where λ is the 1-norm of the Hamiltonian.

“This produced an algorithm with a scaling of O ( M 1 . 5 λ ), where λ is the 1-norm of the Hamiltonian.”

• cites Berry et al., 2019bNot checked yet

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These methods are typically assumed to require fault-tolerance.

“As a result, these methods are typically assumed to require fault-tolerance.”

• AssumptionThe authors state this as an assumption: The methods discussed above require fault-tolerance.

A time dependent Taylor series approach in the interaction picture scales as O(M^2) in a plane wave dual basis.

“A time dependent form of the Taylor series approach was used for simulation in the 'interaction picture', which enables more efficient time evolution in a plane wave dual basis, scaling as O ( M 2 ) (Low and Wiebe, 2018) (although this algorithm uses O ( M log 2 ( M )) qubits).”

• cites Low and Wiebe, 2018Not checked yet

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All of the methods discussed above require circuits with a large number of gates.

“Despite this progress, all of the methods discussed above require circuits with a large number of gates.”

The long gate sequences required by these algorithms make them unsuitable for near-term hardware.

“As near-term quantum computers will not have enough physical qubits for error correction, the long gate sequences required by these algorithms make them unsuitable for near-term hardware.”

• AssumptionRests on an assumption: Near-term quantum computers will not have enough physical qubits for error correction.

The coupled cluster (CC) method can be extended to produce the unitary coupled cluster (UCC) ansatz.

“Most notably, the coupled cluster (CC) method discussed in Sec. III.C.4 can be extended to produce the unitary coupled cluster (UCC) ansatz (Bartlett et al. , 1989; Hoffmann and Simons, 1988).”

• cites Bartlett et al., 1989Not checked yet

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• cites Hoffmann and Simons, 1988Not checked yet

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Haah et al. (2018) obtained a simulation algorithm requiring O(Mt) gates to simulate time evolution for time t under lattice Hamiltonians.

“Haah et al. (2018) made use of arguments about the speed of information propagation in lattice systems to obtain a simulation algorithm requiring O ( Mt ) gates to simulate time evolution for time t under lattice Hamiltonians.”

• cites Haah et al., 2018Not checked yet

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Randomly initialised hardware efficient ansätze are not a scalable solution for problems in quantum computational chemistry.

“This suggests that randomly initialised hardware efficient ans¨ atze are not a scalable solution for problems in quantum computational chemistry.”

Quantum circuits can calculate the analytic gradient of the energy with respect to one of the parameters, avoiding finite difference formulae.

“Quantum circuits have been proposed to calculate the analytic gradient of the energy with respect to one of the parameters (Guerreschi and Smelyanskiy, 2017; Mitarai and Fujii, 2018; Schuld et al. , 2019). This avoids the use of finite difference formulae, which restrict the accuracy of gradient evaluation, as the finite difference considered is limited by the noise in the energy evaluation.”

• cites Guerreschi and Smelyanskiy, 2017Not checked yet

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• cites Mitarai and Fujii, 2018Not checked yet

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• cites Schuld et al., 2019Not checked yet

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Around 10^13 samples would be required per energy evaluation for a 112 spin-orbital molecule such as Fe2S2.

“For example, Wecker et al. (2015a) found that around 10 13 samples would be required per energy evaluation for a 112 spin-orbital molecule such as Fe 2 S 2 .”

• cites Wecker et al. (2015a)Not checked yet

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The number of measurements required to estimate the energy to a precision ε is bounded by a formula involving the coefficients of each Pauli string in the Hamiltonian.

“McClean et al. (2016) and Romero et al. (2019) showed that the number of measurements N m , required to estimate the energy to a precision ϵ , is bounded by”

• cites McClean et al. (2016)Not checked yet

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• cites Romero et al. (2019)Not checked yet

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For LiH in an STO-3G basis, taking into account symmetries and a reduced active space allows accurate results while considering only around 12 excitation operators, compared to around 200 naively.

“For example, the LiH molecule in an STO-3G basis naively has around 200 excitation operators to consider. However, taking into account symmetries and a reduced active space, one can achieve accurate results while considering only around 12 excitation operators (Hempel et al. , 2018).”

• cites Hempel et al., 2018Not checked yet

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The Hamiltonian variational ansatz appears particularly suitable for the Fermi-Hubbard model, which only has O(M) terms in its Hamiltonian.

“The Hamiltonian variational ansatz appears particularly suitable for the Fermi-Hubbard model, which only has O ( M ) terms in its Hamiltonian.”

Canonical gradient descent based methods have struggled to find the ground state, due to the high levels of noise present in current quantum devices.

“In contrast, canonical gradient descent based methods have struggled to find the ground state, due to the high levels of noise present in current quantum devices (Peruzzo et al. , 2014).”

• cites Peruzzo et al., 2014Not checked yet

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It is possible to implement Trotter steps of the Fermi-Hubbard Hamiltonian with O(√M) depth and O(M^1.5) gates on a linear array of qubits with nearest-neighbour connectivity.

“Kivlichan et al. (2018) showed that it is possible to implement Trotter steps of the Fermi-Hubbard Hamiltonian with O ( √ M ) depth and O ( M 1 . 5 ) gates on a linear array of qubits with nearest-neighbour connectivity.”

• cites Kivlichan et al. (2018)Not checked yet

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In comparisons of out-of-the-box optimisers, the Nelder-Mead simplex algorithm utilised in experimental studies was outperformed by both other direct search approaches and gradient based approaches, such as LBFGS-B.

“In comparisons of 'out-of-the-box' optimisers (McClean et al. , 2016; Romero et al. , 2019), the NelderMead simplex algorithm utilised in experimental studies was outperformed by both other direct search approaches, and gradient based approaches, such as LBFGS-B.”

• cites McClean et al., 2016Not checked yet

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• cites Romero et al., 2019Not checked yet

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The UCCSD gate count has been improved to O(M^3)-O(M^4) gates using a low-rank decomposition of the UCCSD tensor.

“or O ( M 3 ) - O ( M 4 ) gates using a low-rank decomposition of the UCCSD tensor (Motta et al. , 2018).”

• cites Motta et al., 2018Not checked yet

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• AssumptionThe authors state this as an assumption: A single Trotter step can be used to implement the UCC operator.

The efficiency of the Hamiltonian variational ansatz is determined by the number of terms in the Hamiltonian.

“The efficiency of this ansatz is determined by the number of terms in the Hamiltonian.”

If only the energy expectation value is of interest, it is possible to use low rank Hamiltonian decomposition and orbital basis rotations to divide the Hamiltonian into O(M) measurement groups.

“If we are only interested in measuring the energy expectation value, rather than the 1 and 2-RDMs, Huggins et al. (2019b) showed that it is possible to use the low rank Hamiltonian decomposition technique (discussed in Sec. IV.C.2) and orbital basis rotations to divide the Hamiltonian into O ( M ) measurement groups.”

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The Hamiltonian variational ansatz leads to a scaling of approximately O(M^4) when considering a Gaussian molecular orbital basis.

“This leads to a scaling of approximately O ( M 4 ) when considering a Gaussian molecular orbital basis (Wecker et al. , 2015a).”

• cites Wecker et al., 2015aNot checked yet

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The number of parameters in the Hamiltonian variational ansatz scales linearly with the number of Trotter steps, S.

“The number of parameters in this ansatz scales linearly with the number of Trotter steps, S .”

The electronic structure Hamiltonian in a plane wave dual basis only contains O(M^2) terms.

“The electronic structure Hamiltonian in a plane wave dual basis only contains O ( M 2 ) terms.”

All methods were liable to becoming trapped in local minima, even for small systems.

“All methods were liable to becoming trapped in local minima, even for small systems.”

• cites McClean et al., 2016Not checked yet

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• cites Romero et al., 2019Not checked yet

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When taking into account covariances between commuting terms, the total number of measurements required for small molecules was reduced by several orders of magnitude.

“When taking into account covariances between commuting terms, the total number of measurements required for small molecules was reduced by several orders of magnitude.”

In practice, the gate scaling of UCCSD is typically better than the bounds given above, as many excitations are forbidden by the symmetry point groups of molecular orbitals.

“In practice, the gate scaling is typically better than the bounds given above, as many excitations are forbidden by the symmetry point groups of molecular orbitals.”

By grouping terms at a fermionic level, the fermionic 2-RDM can be measured using O(M^2) circuits, with an additional gate depth of O(M).

“By grouping terms at a fermionic level, rather than a qubit level, Bonet-Monroig et al. (2019) devised a method for measuring the fermionic 2-RDM using O ( M 2 ) circuits (an additional gate depth of O ( M ) is required).”

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In general, error mitigation techniques only require a multiplicative overhead in the number of measurements required, if the error rate is sufficiently low.

“In general, these techniques only require a multiplicative overhead in the number of measurements required, if the error rate is sufficiently low.”

QSE has been experimentally demonstrated using a two qubit superconducting system to measure the ground and excited state energies of H2.

“The QSE has been experimentally demonstrated, using a two qubit superconducting system to measure the ground and excited state energies of H 2 (Colless et al. , 2018).”

• cites Colless et al. , 2018Not checked yet

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Using a combination of process and gate set tomography reduces the effect of errors due to state preparation and measurement.

“The latter approach reduces the effect of errors due to state preparation and measurement (Endo et al. , 2017).”

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Overlap based methods exploit the orthogonality of energy eigenstates to find other eigenstates by ensuring they are orthogonal to the original state.

“Overlap based methods exploit the orthogonality of energy eigenstates. Once an eigenstate is found, we can find other eigenstates by ensuring that they are orthogonal to the original state (Endo et al. , 2018a; Higgott et al. , 2018).”

• cites Endo et al., 2018aNot checked yet

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• cites Higgott et al., 2018Not checked yet

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Probabilistic error cancellation works by effectively realising the inverse of an error channel N^-1 such that N^-1(N(ρ0)) = ρ0.

“The probabilistic error cancellation method introduced by Temme et al. (2017) works by effectively realising the inverse of an error channel, N -1 , such that N -1 ( N ( ρ 0 )) = ρ 0 .”

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In practice, it is not possible to exactly measure all possible terms resulting from errors if there are many gates in the circuit.

“In practice, it is not possible to exactly measure all of the possible terms resulting from errors if there are many gates in the circuit.”

The original QSE formulation used a polynomial number of additional measurements to find the excited states of a quantum system.

“The original formulation of the QSE used a polynomial number of additional measurements to find the excited states of a quantum system (McClean et al. , 2017a).”

• cites McClean et al., 2017aNot checked yet

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Probabilistic error cancellation requires full knowledge of the noise model associated with each gate.

“Probabilistic error cancellation requires full knowledge of the noise model associated with each gate.”

The VQE can be inherently robust to some coherent errors, such as qubit over-rotation.

“It has been shown both theoretically (McClean et al. , 2016) and experimentally (O'Malley et al. , 2016) that the VQE can be inherently robust to some coherent errors, such as qubit over-rotation.”

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• cites O'Malley et al., 2016Not checked yet

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After finding the ground state of a Hamiltonian H with VQE, replacing the Hamiltonian with H' = H + α|E0><E0| makes the ground state of H' the first excited state of H.

“The ground state of the updated Hamiltonian H ′ is no longer | E 0 〉 , but the first excited state | E 1 〉 of the original Hamiltonian H .”

Small experimental demonstrations of the VQE have shown that noise can still prevent reaching the desired levels of accuracy.

“However, small experimental demonstrations of the VQE have shown that noise can still prevent us from reaching the desired levels of accuracy (Hempel et al. , 2018; Kandala et al. , 2017).”

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• cites Kandala et al., 2017Not checked yet

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The QSE single excitation linear response subspace is designed to target the low-lying excited states.

“This is designed to target the low-lying excited states, which are assumed to differ from the ground state by a small number of excitations.”

• AssumptionThe authors state this as an assumption: Low-lying excited states differ from the ground state by a small number of excitations.

The variance in the estimate of the noiseless measurement result O0 is increased by a factor γG, where γ is the overhead coefficient and G is the number of gates in the circuit.

“The variance in our estimate of ¯ O 0 is increased by a factor of γ G , where γ is the overhead coefficient, and G is the number of gates in the circuit.”

It is possible to mitigate some errors by using symmetry checks on a suitably constructed ansatz state.

“It is also possible to mitigate some errors by using symmetry checks on a suitably constructed ansatz state (Bonet-Monroig et al. , 2018; McArdle et al. , 2018c).”

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• cites McArdle et al. , 2018cNot checked yet

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The QSE can be used to recover the energy contribution of virtual orbitals without requiring additional qubits to represent the virtual orbitals.

“Takeshita et al. (2019) showed how the QSE can be used to recover the energy contribution of virtual orbitals, without requiring additional qubits to represent the virtual orbitals.”

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If errors are not dealt with, they will corrupt the results of algorithms, rendering calculations meaningless.

“If these errors are not dealt with, they will corrupt the results of our algorithms, rendering the calculations meaningless.”

Probabilistic error cancellation has been shown to work for general Markovian noise.

“The probabilistic error cancellation method described above has been shown to work for general Markovian noise (Endo et al. , 2017), and has also been extended to work with temporally correlated errors and low frequency noise (Huo and Li, 2018).”

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The QSE was experimentally demonstrated in a two qubit superconducting system to find the ground and excited states of H2.

“The QSE was experimentally demonstrated in a two qubit superconducting system to find the ground and excited states of H 2 (Colless et al. , 2018).”

• cites Colless et al., 2018Not checked yet

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Barkoutsos et al. introduced a transformation of the Hamiltonian such that it only measures the energy of excitations above the Hartree-Fock state (the correlation energy).

“Barkoutsos et al. (2018) introduced a transformation of the Hamiltonian, such that it only measures the energy of excitations above the Hartree-Fock state (the correlation energy).”

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Exponential extrapolation is a more appropriate extrapolation technique for large quantum circuits.

“Exponential extrapolation was introduced by Endo et al. (2017) as a more appropriate extrapolation technique for large quantum circuits.”

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The extrapolation method has been demonstrated for VQE experiments in molecular chemistry and nuclear physics.

“The extrapolation method has been demonstrated for VQE experiments in both molecular chemistry (Kandala et al. , 2018; Ollitrault et al. , 2019), and nuclear physics (Shehab et al. , 2019).”

• cites Kandala et al., 2018Not checked yet

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• cites Ollitrault et al., 2019Not checked yet

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• cites Shehab et al., 2019Not checked yet

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Numerical simulations showed the impact that noise has on phase estimation based chemistry calculations.

“This was confirmed for the case of chemistry calculations by Sawaya et al. (2016), who used numerical simulations to show the impact that noise has on phase estimation based chemistry calculations.”

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States with electron number far from the true value appear to have a larger energy variance than those with smaller particle number errors.

“A key concern for the VQE is preserving particle number, as states with electron number far from the true value appear to have a larger energy variance than those with smaller particle number errors (Sawaya et al. , 2016).”

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The QSE with single and double excitations can be understood as an approximation to the equation of motion (EoM) method for finding excited states.

“Ollitrault et al. (2019) showed that the QSE with single and double excitations can be understood as an approximation to the 'equation of motion' (EoM) method for finding excited states, which was originally introduced as a classical method in quantum chemistry by Rowe (1968).”

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Quantum computers can store the FCI wavefunction of M spin-orbitals using only M qubits in second quantisation.

“As discussed in Sec. IV.B, quantum computers can store the FCI wavefunction of M spin-orbitals using only M qubits in second quantisation.”

Many problems of scientific interest fit the sweet spot description, including transition metal catalysts and the Fermi-Hubbard model.

“Interestingly, many problems of scientific interest fit this description, including: transition metal catalysts (Podewitz et al. , 2011; Vogiatzis et al. , 2019) and the Fermi-Hubbard model (LeBlanc et al. , 2015).”

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• cites Vogiatzis et al. , 2019Not checked yet

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• cites LeBlanc et al. , 2015Not checked yet

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Coupled cluster methods (often CCSD) are applicable to large systems (hundreds of spin-orbitals) which do not display strong static correlation.

“Coupled cluster methods (often CCSD) are some of the most widely used high-accuracy methods. They are applicable to large systems (hundreds of spinorbitals (Bartlett and Musiaglyph[suppress] l, 2007)) which do not display strong static correlation.”

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Working in a suitably large basis set is crucial for obtaining accurate results.

“This highlights that working in a suitably large basis set is crucial for obtaining accurate results.”

McClean et al. (2019b) developed a related extension of QSE that effectively engineered additional symmetries using an error correcting code.

“A related extension of the quantum subspace expansion was developed by McClean et al. (2019b), who effectively engineered additional symmetries in the system using an error correcting code.”

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All Pauli terms arising from the same excitation operators commute.

“It was shown by Romero et al. (2019) that all Pauli terms arising from the same excitation operators commute.”

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Post-processing the results of individual error reduction produces a more accurate expectation value than would be obtained without the mitigation technique.

“Post-processing the results produces a more accurate expectation value than would be obtained without the mitigation technique.”

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These problems are among the best targets for quantum computers.

“This would imply that these problems are among the best targets for quantum computers.”

There appears to be an untreated 'sweet spot' of systems with around 100 to 200 spin-orbitals that require high accuracy calculations.

“We can see from the above discussion that there appears to be an untreated 'sweet spot', of systems with around 100 to 200 spin-orbitals, that require high accuracy calculations.”

The ground state of the H2 molecule at its equilibrium bond distance is given by the wavefunction in Eq. (98) (Helgaker et al., 2014).

“and the ground state of the H 2 molecule at its equilibrium bond distance is given by (Helgaker et al. , 2014)”

• cites Helgaker et al. , 2014Not checked yet

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Over 95% of the approved drug molecules in DrugBank 5.0 are larger than the 100-200 spin-orbital systems that might be simulated with a small, error-corrected quantum computer.

“For example, Yamazaki et al. (2018) noted that over 95 % of the approved drug molecules in DrugBank 5.0 are larger than these 100-200 spin-orbital systems that we might aim to simulate with a small, error-corrected quantum computer.”

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Tubman et al. (2018) presented an algorithm which can prepare a suitable initial state by leveraging a classical adaptive configuration interaction method.

“Tubman et al. (2018) presented an algorithm which can prepare a suitable initial state by leveraging a classical adaptive configuration interaction method.”

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This estimate was subsequently reduced through a series of algorithmic optimisations.

“This estimate was subsequently reduced through a series of algorithmic optimisations, as described in Sec V.A.4.a.”

The plane wave dual basis can reduce the costs of simulations.

“This enables the use of the plane wave dual basis (Sec. III.D.3), which as discussed in Sec. V.A.4.b can reduce the costs of simulations.”

Berry et al. (2019b) carried out a similar resource estimation using the more accurate FeMo-co active space of Li et al. (2018), with 113 electrons in 152 spin-orbitals, using an algorithm based on qubitization and low-rank decompositions of the Hamiltonian.

“Asimilar resource estimation (although using the more accurate FeMo-co active space of Li et al. (2018), with 113 electrons in 152 spin-orbitals) was carried out by Berry et al. (2019b), who used the algorithm based on qubitization and low-rank decompositions of the Hamiltonian, described in Sec. V.A.4.”

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• cites Li et al. (2018)Not checked yet

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These improvements, combined with the algorithmic advances described throughout this review, have contributed to a significant reduction in the resources required for chemistry calculations, compared to the initial estimates.

“These improvements, combined with the algorithmic advances described throughout this review, have contributed to a significant reduction in the resources required for chemistry calculations, compared to the initial estimates.”

Reiher et al. (2017) found that this would require around 10^14 T gates.

“They found that this would require around 10 14 T gates.”

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Berry et al. (2019b) showed that the cost of Toffoli gate distillation is equivalent to around 1 million physical qubits, working for 2 months (assuming 10^-3 error rates and surface code cycle times of 1 μs).

“While a complete fault-tolerant resource analysis for this new approach has not yet been performed, those authors showed that the cost of Toffoli gate distillation is equivalent to around 1 million physical qubits, working for 2 months (assuming 10 -3 error rates and surface code cycle times of 1 µ s), at our time of writing.”

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Kivlichan et al. (2019a) found that simulations of a 100 site Fermi-Hubbard model, and a 128 spin-orbital homogeneous electron gas would both require on the order of 10^6 Toffoli gates and 10^7-10^8 T gates.

“They found that simulations of a 100 site Fermi-Hubbard model, and a 128 spin-orbital homogeneous electron gas would both require on the order of 10 6 Toffoli gates and 10 7 -10 8 T gates.”

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This approach reduced the resources required to around 10^11 Toffoli gates.

“This approach reduced the resources required to around 10 11 Toffoli gates (which are currently the bottleneck for this approach).”

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The length of circuit that we can implement without error correction is heavily limited by noise.

“However, the length of circuit that we can implement without error correction is heavily limited by noise.”

Kivlichan et al. (2019a) performed resource estimations for condensed matter and solid state problems, using the Trotter based algorithm.

“Kivlichan et al. (2019a) also performed resource estimations for condensed matter and solid state problems, using the Trotter based algorithm discussed in Sec. V.A.4.b.”

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The algorithm investigated in Jones et al. (2012) (the algorithm of Kassal et al. (2008)) has since been surpassed by the algorithm of Kivlichan et al. (2017).

“The algorithm investigated in that work (the algorithm of Kassal et al. (2008)) has since been surpassed by the algorithm of Kivlichan et al. (2017).”

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• cites Kivlichan et al. (2017)Not checked yet

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Subsequent improvements in fault-tolerance protocols have further reduced these physical resources.

“Subsequent improvements in faulttolerance protocols have further reduced these physical resources (Fowler and Gidney, 2018; Gidney and Fowler, 2019; Litinski, 2019a,b).”

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• cites Gidney and Fowler, 2019Not checked yet

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• cites Litinski, 2019a,bNot checked yet

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This reduced the necessity of having a large overlap with the ground state.

“This reduced the necessity of having a large overlap with the ground state.”

In particular, materials in the condensed phase and Fermi-Hubbard simulations so far require considerably fewer resources than simulations of individual molecules.

“In particular, materials in the condensed phase and Fermi-Hubbard simulations so far require considerably fewer resources than simulations of individual molecules.”

Considerable work to reduce the cost of operations in the surface code (including magic state distillation) has reduced the overhead of error correction by several orders of magnitude.

“There has been considerable work to reduce the cost of operations in the surface code (including magic state distillation), which has reduced the overhead of error correction by several orders of magnitude (Fowler and Gidney, 2018; Gidney and Fowler, 2019; Litinski, 2019a,b; Trout and Brown, 2015).”

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• cites Gidney and Fowler, 2019Not checked yet

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• cites Litinski, 2019a,bNot checked yet

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As such, the Jones et al. (2012) work is not directly comparable to the methods described above.

“As such, it is not directly comparable to the methods described above.”

It may be many years before we possess a quantum computer with the resources required to implement these algorithms, given the challenges in scaling up hardware and performing quantum error correction.

“It may be many years before we possess a quantum computer with the resources required to implement these algorithms, given the challenges in scaling up hardware and performing quantum error correction (Gambetta et al. , 2017).”

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• AssumptionRests on an assumption: Current quantum computers possess only around 100 physical qubits, and a fully protected logical qubit has not yet been demonstrated.

Kivlichan et al. (2019a) found that if an intensive (e.g. absolute) error in the energy is targeted, then their algorithm was less efficient than the qubitization algorithm of Babbush et al. (2018b).

“Kivlichan et al. (2019a) found that if an intensive (e.g. absolute) error in the energy is targeted, then their algorithm was less efficient than the qubitization algorithm of Babbush et al. (2018b).”

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These results suggest that certain calculations with around 100 spin-orbitals may be better suited to early quantum computers than others.

“These results suggest that certain calculations with around 100 spin-orbitals may be better suited to early quantum computers than others.”

Assuming error rates of 10^-3, 1 μs to implement a T gate (including surface code decoding), and 10 μs feedforward, the Babbush et al. (2018b) estimates led to resources of around 2-3 million qubits, running for tens of hours.

“Assuming error rates of 10 -3 , 1 µ s to implement a T gate (including surface code decoding), and 10 µ s feedforward, this led to resources of around 2-3 million qubits, running for tens of hours (using the best fault-tolerance protocols available when that work was completed).”

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Algorithm complexities are measured in terms of the number of T and/or Toffoli gates that they contain, as these are often the dominant contribution to the cost of the algorithm.

“As a result, algorithm complexities are measured in terms of the number of T and/or Toffoli gates that they contain, as these are often the dominant contribution to the cost of the algorithm.”

Assuming error rates of 10^-3, surface code error detection times of 1 μs, and surface code error decoding times of 10 μs, the Kivlichan et al. (2019a) estimates require around 400,000-600,000 physical qubits, running for a couple of hours.

“Assuming error rates of 10 -3 , surface code error detection times of 1 µ s, and surface code error decoding times of 10 µ s, they require around 400,000 - 600,000 physical qubits, running for a couple of hours.”

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Reiher et al. (2017) calculated the resources required to perform an FCI calculation on an active space of 54 electrons in 108 spinorbitals for FeMo-co, using a Trotter-based approach to phase estimation.

“Reiher et al. (2017) carried out a fault-tolerant resource estimation for the problem of biological dinitrogen fixation, as described in Sec. III.A.1. Those authors calculated the resources required to perform an FCI calculation on an active space of 54 electrons in 108 spinorbitals for FeMo-co, using a Trotter-based approach to phase estimation.”

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The dominant Slater determinants in the wavefunction typically converge much more quickly than the correlation energy.

“This stems from the fact that the dominant Slater determinants in the wavefunction typically converge much more quickly than the correlation energy.”

This technique can also be combined with that of Berry et al. (2018), which used classically obtained knowledge of the energy eigenvalues to reduce the number of times that phase estimation must be repeated.

“This technique can also be combined with that of Berry et al. (2018), which used classically obtained knowledge of the energy eigenvalues to reduce the number of times that phase estimation must be repeated.”

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This algorithm was numerically shown to provide good estimates for the ground states of many systems of interest in chemistry, physics and materials science.

“This algorithm was numerically shown to provide good estimates for the ground states of many systems of interest in chemistry, physics and materials science.”

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Because these simulations consider a loose-but-rigorous bound on the energy error, these resource estimates may be overly pessimistic.

“As these simulations consider a loose-but-rigorous bound on the energy error (as discussed in Sec. V.A.5), these resource estimates may be overly pessimistic.”

It still requires on the order of 100,000 physical qubits to surpass classical techniques.

“Despite these promising results, and recent, rapid improvements, we see that it still requires on the order of 100 , 000 physical qubits to surpass classical techniques.”

• Also stated in: § C. Quantum resources: near to medium-term · p. 49 · hedged

  “As discussed above, existing estimates for surpassing classical calculations require on the order of 100,000 physical qubits, in order to implement quantum error correction.”

This calculation showed improved accuracy over an un-encoded calculation, due to the use of post-selection.

“This calculation showed improved accuracy over an un-encoded calculation, due to the use of post-selection.”

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This is comparable to the T gate counts required for phase estimation based approaches described in Sec. VIII.B.

“This is comparable to the T gate counts required for phase estimation based approaches described in Sec. VIII.B.”

The total algorithm scaling would be around 40,000 two-qubit gates for a 10 × 10 site Fermi-Hubbard model.

“which is around 40,000 two-qubit gates for a 10 × 10 site Fermi-Hubbard model (which requires 200 qubits).”

An initial foray into error corrected VQE was conducted by Urbanek et al. (2019), who experimentally implemented a VQE calculation on the H2 molecule encoded in the [[4,2,2]] error detecting code.

“An initial foray into this area was conducted by Urbanek et al. (2019), who experimentally implemented a VQE calculation on the H 2 molecule encoded in the [[4,2,2]] error detecting code.”

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Our estimate of 40,000 two-qubit gates is approximately equal to the number of gates that we were limited to by noise in our back-of-the-envelope calculation above.

“Our estimate of 40,000 two-qubit gates is approximately equal to the number of gates that we were limited to by noise in our back-of-the-envelope calculation above.”

The phase estimation based algorithm of Kivlichan et al. (2019a) is already only a constant factor less efficient (in terms of T/Toffoli gates) than just implementing time evolution under the Fermi-Hubbard Hamiltonian directly.

“As an example, the phase estimation based algorithm of Kivlichan et al. (2019a) is already only a constant factor less efficient (in terms of T/Toffoli gates) than just implementing time evolution under the Fermi-Hubbard Hamiltonian directly (as is required for a Hamiltonian variational ansatz).”

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Previous work has shown that the Hamiltonian variational ansatz performs well for the Fermi-Hubbard model.

“Previous work has shown that the Hamiltonian variational ansatz performs well for the Fermi-Hubbard model, although it is not yet known how many Trotter steps may be required for accurate results.”

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• cites Reiner et al. , 2018aNot checked yet

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Wecker et al. (2015a) achieved good convergence for a 12 site problem with 20 Trotter steps.

“Wecker et al. (2015a) achieved good convergence for a 12 site problem with 20 Trotter steps.”

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For the hypothetical O(M^2) scaling variational algorithm described above, we may still need around 4 × 10^5 to 4 × 10^6 T gates, neglecting constant factors.

“For the hypothetical O ( M 2 ) scaling variational algorithm described above, we may still need around 4 × 10 5 to 4 × 10 6 T gates, neglecting constant factors.”

• AssumptionRests on an assumption: Synthesizing an arbitrary angle rotation gate can require at least 10 to 100 T gates.

A 10 × 10 site Fermi-Hubbard model requires 200 qubits.

“which is around 40,000 two-qubit gates for a 10 × 10 site Fermi-Hubbard model (which requires 200 qubits).”

The G1 set is a small set of molecules whose energy is known extremely accurately.

“This is a small set of molecules whose energy is known extremely accurately.”

• cites Pople et al. , 1989Not checked yet

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If the number of Trotter steps required to find the ground state scaled only linearly with the number of spin-orbitals, the total algorithm scaling would be O(M^2).

“Nevertheless, even if the number of Trotter steps required to find the ground state scaled only linearly with the number of spin-orbitals, the total algorithm scaling would be O ( M 2 ), which is around 40,000 two-qubit gates for a 10 × 10 site Fermi-Hubbard model (which requires 200 qubits).”