Quantum computational chemistry

In the paper

“In total, there are P 3 N × 2 N complex amplitudes, showing that the memory required scales exponentially with the size of the simulated system.”

In the paper

“For LiH in the STO-3G basis, we consider { 1 s, 2 s, 2 p x , 2 p y , 2 p z } functions for lithium, and a single { 1 s } orbital for hydrogen. This gives a total of 12 spin-orbitals. We can reduce this problem to one of six qubits, as illustrated in Fig. 12.”

In the paper

“The lowest energy computational basis state of ccPVDZ H 2 in the Jordan-Wigner encoding (18 qubits) is | 000000001000000001 〉 . The corresponding BK-tree mapped state (16 qubits) is given by | 0001011100010111 〉 .”

In the paper

“Our final Hamiltonian acts on 6 qubits, but differs in energy from the full 12 qubit Hamiltonian by only 0 . 2 mHartree.”

In the paper

“The reader will find that the 9 th and 18 th orbitals store the number of spin up electrons and total number of electrons, respectively. As a result, they can be removed. This reduces the problem to one of 16 qubits.”

In the paper

“The 1-RDM in the canonical orbital basis from a CISD calculation on LiH (at an internuclear separation of 1.45 ˙ A) is given by”

In the paper

“The JW mapped HF state (8 qubits) is given by | 00010001 〉 . Using the matrix given above, the BK mapped HF state (8 qubits) is | 00111011 〉 . When the two conserved qubits are removed, the BK mapped HF state (6 qubits) is | 011011 〉 .”

In the paper

“Finally, most of the prior work in quantum computational chemistry has focused on this problem.”

In the paper

“However, we stress that to achieve this ultimate goal we would need considerable further developments - both in the technology required to build such as powerful quantum computer, and the theory behind an appropriate algorithm and model.”

In the paper

“Despite ongoing experimental efforts, no group has yet demonstrated a single fully error corrected qubit.”

In the paper

“Therefore any multi-qubit observable can be expanded into strings of Pauli operators, the expectation values of which we can measure efficiently with a quantum computer.”

In the paper

“Knowledge of the energy eigenstates enables the prediction of reaction rates, location of stable structures, and determination of optical properties (Helgaker et al. , 2012).”

What the sources say

• Not checked yetcites Helgaker et al., 2012

In the paper

“Despite being a relatively young field, quantum computational chemistry has grown extremely rapidly, and has already evolved beyond the stage that it can be fully described by a single review.”

In the paper

“Secondly, the machinery developed to solve this problem on quantum computers is easily applied to other types of problems, such as finding transition states, or understanding the vibrational structure of molecules.”

In the paper

“Moreover, the gate counts currently estimated for transformative chemistry simulations likely signal the need for quantum error correction, which requires orders of magnitude more qubits, and lower error rates, than are currently available (Babbush et al. , 2018b; Kivlichan et al. , 2019a; Mueck, 2015).”

What the sources say

• Not checked yetcites Babbush et al., 2018b
• Not checked yetcites Kivlichan et al., 2019a
• Not checked yetcites Mueck, 2015

In the paper

“Experimental developments have accompanied these theoretical milestones, with many groups demonstrating proof of principle chemistry calculations.”

In the paper

“However, limited by hardware capabilities, these experiments focus only on small chemical systems that we are already able to simulate classically.”

In the paper

“The Pauli operators and identity matrix, multiplied by real coefficients, form a complete basis for any Hermitian single or multi-qubit operator.”

In the paper

“The runtime and physical resources required by these algorithms are expected to scale polynomially with both the size of the system simulated and the accuracy required.”

In the paper

“Feynman proposed a solution to this problem; using quantum hardware as the simulation platform”

What the sources say

• Not checked yetcites Feynman, 1982

In the paper

“As a result, the circuit model of quantum computing typically decomposes all algorithms into single and two qubit gates.”

In the paper

“To date, several efficient quantum algorithms have been proposed to solve problems in chemistry.”

In the paper

“As a consequence, first quantised basis set methods are rarely, if ever, used in classical computational chemistry calculations.”

In the paper

“The electronic structure problem has received significant attention since it was first introduced in the context of quantum computational chemistry by Aspuru-Guzik et al. (2005), and is widely considered to be one of the first applications of quantum computing.”

What the sources say

• Not checked yetcites Aspuru-Guzik et al., 2005

In the paper

“As such, it may be better to treat the nuclei and electrons on an equal footing, which is best achieved with grid based methods.”

In the paper

“While classical computational chemistry has made tremendous progress in describing and predicting the properties of a multitude of systems, there are some systems that appear classically intractable to simulate with currently known techniques.”

In the paper

“Grid based methods are useful when considering chemical reaction dynamics, or when simulating systems for which the Born-Oppenheimer approximation is not appropriate.”

In the paper

“As we will see in Sec. III.B.2, the information encoded by a Slater determinant can be compressed by moving to the second quantised formalism.”

In the paper

“Nevertheless, quantum computers with more than around 50 qubits are considered too large to exactly simulate classically, and may thus be capable of solving problems which are intractable on even the largest classical supercomputers (Arute et al. , 2019).”

What the sources say

• Not checked yetcites Arute et al., 2019

In the paper

“However, a variational circuit with set parameters is tailored to the time evolution of one or more specific initial states, in contrast to a Trotter circuit, which can be used to time evolve any valid initial state.”

In the paper

“Driven by this potential utility, quantum computational chemistry is rapidly emerging as an interdisciplinary field requiring knowledge of both quantum computing and computational chemistry.”

In the paper

“Nevertheless, quantum computers with more than around 50 qubits are considered too large to exactly simulate classically, and may thus be capable of solving problems which are intractable on even the largest classical supercomputers (Arute et al. , 2019).”

What the sources say

• Not checked yetcites Arute et al., 2019

In the paper

“As a consequence of the Pauli exclusion principle, the electronic wavefunction must be antisymmetric under the exchange of any two electrons.”

In the paper

“It is widely believed that this problem should be efficiently solvable with a quantum computer, for physically relevant systems (Whitfield et al. , 2013).”

What the sources say

• Not checked yetcites Whitfield et al., 2013

In the paper

“While the number of spin-orbitals considered, M , is typically larger than the number of electrons in the system, N , the electrons can only occupy N of the spin-orbitals in a given Slater determinant.”

In the paper

“In particular, Abrams and Lloyd (1999) showed that the phase estimation algorithm (Kitaev, 1995) can be used to find the energy of a quantum system, and collapse it into the desired energy eigenstate.”

What the sources say

• Not checked yetcites Abrams and Lloyd, 1999
• Not checked yetcites Kitaev, 1995

In the paper

“However, building a sufficiently large quantum computer will be a difficult scientific challenge.”

In the paper

“It is important to note that, while it is widely believed to be true, it has not yet been mathematically proven that BPP = BQP (where BPP is the complexity class containing problems that are solvable in polynomial time on a probabilistic classical computer).”

In the paper

“Even if the individual energy values cannot be measured to within chemical accuracy, there is often a fortuitous cancellation of errors, which leads to the energy difference being found to chemical accuracy.”

In the paper

“Computational models of FeMo-co have been proposed which are beyond the reach of current classical methods for solving strongly correlated systems (see Sec. VIII.A) but would be accessible with a small error corrected quantum computer (Berry et al. , 2019b; Reiher et al. , 2017).”

What the sources say

• Not checked yetcites Berry et al., 2019b
• Not checked yetcites Reiher et al., 2017

In the paper

“Interestingly, this is the same regime that is believed to be most relevant to understanding cuprate superconductors (LeBlanc et al. , 2015).”

What the sources say

• Not checked yetcites LeBlanc et al., 2015

In the paper

“However, one key difference is that using a basis set is known as a 'Galerkin discretisation', which ensures that the energy converges to the correct value from above, as the number of basis functions tends to infinity.”

In the paper

“The Haber-Bosch process requires high temperatures and pressures, and is believed to consume up to 2 % of the world's energy output (Reiher et al. , 2017).”

What the sources say

• Not checked yetcites Reiher et al., 2017

In the paper

“Nevertheless, to perform accurate simulations of large chemical systems, we will likely require digital quantum simulation, as it is not yet clear how to protect large analog simulations from errors.”

In the paper

“For example, if we consider using a quantum computer to factor numbers in polynomial time (Shor, 1994), current estimates (at time of writing) suggest that we would require around 20 million physical qubits to factor a number that is too large to tackle using known classical algorithms (Gidney and Eker, 2019).”

What the sources say

• Not checked yetcites Gidney and Eker, 2019

In the paper

“Size consistency is important for obtaining the correct behaviour from calculations to determine reaction rates, as reactants at the beginning of the process are considered non-interacting.”

In the paper

“Secondly, first quantised basis set approaches have found use in quantum computational chemistry, as we will discuss in Sec. IV.A.2.”

In the paper

“Static correlation can arise because the wavefunction may require several determinants to be coupled in a proper spin configuration, or during bond breaking in order to account for the separation of the electrons onto the products (the latter case is often referred to as non-dynamic, or left-right correlation) (Lyakh et al. , 2012).”

What the sources say

• Not checked yetcites Lyakh et al. , 2012

In the paper

“The CI method is effective at recovering dynamic correlation, but less effective at recovering static correlation (Helgaker et al. , 2014).”

What the sources say

• Not checked yetcites Helgaker et al. , 2014

In the paper

“As these single-particle molecular orbitals are chosen such that they diagonalise the non-interacting (or mean-field) Hamiltonian, energy eigenstates can be formed by taking tensor products of each electron in a different molecular spin-orbital.”

In the paper

“Although we will discuss orbital basis sets in more detail in Sec. III.D, we note here that the HF orbitals are not suitable for describing the virtual orbitals that electrons are excited into. This is because the HF method only optimises the occupied orbitals in the single Slater determinant wavefunction.”

In the paper

“The Slater determinant generated from a HF calculation is typically taken as the reference state for post-HF methods, such as configuration interaction and coupled cluster, which seek to capture some of the dynamic correlation energy by including additional determinants, describing excitations above the HF state.”

In the paper

“The MCSCF approach considers a wavefunction with several Slater determinants, and variationally optimises both the molecular orbitals, and the determinant amplitudes simultaneously (Roos et al. , 1980).”

What the sources say

• Not checked yetcites Roos et al. , 1980

In the paper

“Systems with strong static correlation are defined as those where multiple Slater determinants are equally dominant.”

In the paper

“We then assume that each electron moves in the average charge distribution of all of the other electrons, which introduces an effective potential.”

In the paper

“These include excited states (Lischka et al. , 2018), transition states (Szalay et al. , 2012), systems at the dissociation limit (Lyakh et al. , 2012), and transition metals (Vogiatzis et al. , 2019).”

What the sources say

• Not checked yetcites Lischka et al. , 2018
• Not checked yetcites Szalay et al. , 2012
• Not checked yetcites Lyakh et al. , 2012

In the paper

“As considering all determinants is classically intractable, the CI method is typically limited to including a small number of excitations above the reference state: single excitations (CIS), double excitations (CISD), and occasionally triple excitations (CISDT).”

In the paper

“For more accurate results, the CCSD(T) method can be used, which treats the triple excitations pertubatively, and scales in time approximately as O ( M 7 ).”

In the paper

“However, this wavefunction contains a number of determinants which scales exponentially with the number of electrons, making large calculations classically intractable.”

In the paper

“To the best of our knowledge, second quantised grid based methods have only been discussed by Babbush et al. (2018c) in their Appendix A.”

What the sources say

• Not checked yetcites Babbush et al. (2018c)

In the paper

“In order to obey the Pauli exclusion principle, these tensor products must be correctly antisymmetrised.”

In the paper

“However, while the methods described above are often effective at recovering the static correlation energy, they struggle to recover the dynamic correlation energy.”

In the paper

“The RayleighRitz variational principle states that the energy expectation value of a parametrized wavefunction is greater than or equal to the lowest energy eigenvalue of the Hamiltonian being measured.”

In the paper

“If all ( M N ) determinants are included, the wavefunction is known as the full configuration interaction (FCI) wavefunction.”

In the paper

“The Hartree-Fock (HF) method is a mean-field technique which aims to find the dominant Slater determinant in the system wavefunction. This is achieved by optimising the spatial form of the spin-orbitals to minimise the energy of the wavefunction.”

In the paper

“This can be achieved by creating Slater determinants from the molecular orbitals, as described by Eq. (21).”

In the paper

“The canonical implementation of CCSD does not store the wavefunction, as this would be exponentially costly (as the CCSD wavefunction has support on all possible Slater determinants). Instead, coupled non-linear equations can be derived. The solution to these equations is the CCSD approximation to the ground state (Helgaker et al. , 2014; Purvis and Bartlett, 1982).”

What the sources say

• Not checked yetcites Helgaker et al. , 2014
• Not checked yetcites Purvis and Bartlett, 1982

In the paper

“We note that in practice, the molecular orbitals obtained by diagonalising the non-interacting part of the Hamiltonian will likely form a poor basis for the system.”

In the paper

“In the HF method, we first neglect the Coulomb repulsion term in the electronic structure Hamiltonian (Eq. (17)), reducing the problem to one of N independent electrons.”

In the paper

“This Hamiltonian has only O ( M ) terms, where M is the number of spin-sites.”

In the paper

“Dynamic correlation is a typically small correction (Hattig et al. , 2012), arising from the Coulomb repulsion between electrons.”

What the sources say

• Not checked yetcites Hattig et al. , 2012

In the paper

“These two metrics are closely related, and while either can be used, the latter produces more compact basis sets, and hence is used more frequently in practice (Helgaker et al. , 2014).”

What the sources say

• Not checked yetcites Helgaker et al. , 2014

In the paper

“Observables must be independent of the representation used.”

In the paper

“However, as a mean-field solution, the HF method neglects the effects of dynamic and static correlation in the wavefunction.”

In the paper

“The importance of each of the virtual atomic orbitals is determined by their contribution to either the electron density (atomic natural orbitals) or the atomic correlation energy (correlation-consistent basis sets).”

What the sources say

• Not checked yetcites Helgaker et al. , 2014

In the paper

“Static correlation occurs when more than one Slater determinant is equally dominant in the wavefunction (Hattig et al. , 2012).”

What the sources say

• Not checked yetcites Hattig et al. , 2012

In the paper

“The coupled cluster (CC) method also includes additional determinants to recover the correlation energy, but uses a product parametrization.”

In the paper

“If all determinants are included, we recover the full configuration interaction (FCI) wavefunction, generated by considering all excitations above the Hartree-Fock wavefunction”

In the paper

“The time taken to solve these equations scales as O (( M -N ) 4 N 2 ) (Purvis and Bartlett, 1982), while the memory needed to store the molecular integrals needed scales as O ( M 4 ).”

What the sources say

• Not checked yetcites Purvis and Bartlett, 1982

In the paper

“The configuration interaction (CI) method generates a correlated wavefunction by considering excitations above a reference state, typically the Hartree-Fock state.”

In the paper

“When all of the excitation operators T i are included, the CC method recovers the full configuration interaction wavefunction - however, performing this calculation would be exponentially costly.”

In the paper

“Kivlichan et al. (2017) also showed that there exist systems where the grid spacing must decrease exponentially with the number of particles in the system to maintain constant accuracy. Consequently, these systems are not efficient to simulate using this method.”

What the sources say

• Not checked yetcites Kivlichan et al. (2017)

In the paper

“Although the Schr¨ odinger equation can be solved exactly for one electron atoms, the orbitals obtained become diffuse too rapidly to accurately describe manyelectron atoms, especially close to the nuclei (Helgaker et al. , 2014).”

What the sources say

• Not checked yetcites Helgaker et al. , 2014

In the paper

“These basis sets can be further improved by adding additional orbitals with higher angular momenta, which make the angular part of the wavefunction more flexible. These orbitals are called 'polarisation functions', as they describe the polarisation of atomic charge caused by bonding (for example, the 6-31G* basis).”

In the paper

“Additional accuracy can be obtained by using cc-PV n Z basis sets (correlation consistent polarised valence n zeta), introduced by Dunning Jr. (1989). These include additional unoccupied ('virtual') orbitals to recover the correlation energy. The virtual orbitals are generated from correlated calculations on atoms.”

What the sources say

• Not checked yetcites Dunning Jr. (1989)

In the paper

“As ρ 1 is a Hermitian operator, we can diagonalise it with a unitary transform. This is a basis change from the canonical orbitals to the 'natural molecular orbitals'. The diagonal elements of the basis transformed ρ 1 are called the natural orbital occupation numbers (NOONs).”

In the paper

“Spin-orbitals with a NOON close to 0 or 1 (compared to the other NOONs) can be assumed to be empty or occupied, respectively.”

In the paper

“Using a grid based method, rather than basis sets, means that the Born-Oppenheimer approximation is not required. As a result, we are able to treat the electrons and nuclei on an equal footing using grid based methods, which can be important for systems undergoing reactions.”

In the paper

“The number of orbitals considered determines the runtime and memory requirements of classical chemistry algorithms. In the case of quantum computational chemistry, the number of basis functions determines the number of qubits and gate operations required to solve the problem”

In the paper

“The simulation of chemical systems using a grid based method can require considerably more qubits than in basis set approaches. For example, it would require 96 logical qubits to store the position of a single spinless particle to 32 bits of accuracy using a grid based approach. This can be contrasted with basis set approaches, where interesting molecules or Fermi-Hubbard models could be simulated”

In the paper

“This reduces the number of Hamiltonian terms from O ( M 4 ) to O ( M 3 ) in the plane wave basis, and O ( M 2 ) in the plane wave dual basis.”

What the sources say

• Not checked yetcites Babbush et al. , 2018c

In the paper

“While the STO functions exhibit many desirable features, they make evaluating the two-electron integrals in Eq. (28) computationally expensive.”

In the paper

“These can be obtained by numerically solving the Schr¨ odinger equation using grid based methods with a very fine grid spacing, which is only possible for small atoms or simple molecules.”

In the paper

“Moreover, the conventional CC method described above is a single determinant reference state method. Consequently, it does not tend to perform well when applied to multireference states, which are required to treat systems with strong static correlation (Lischka et al. , 2018; Lyakh et al. , 2012).”

What the sources say

• Not checked yetcites Lischka et al. , 2018
• Not checked yetcites Lyakh et al. , 2012

In the paper

“These plane wave basis sets are well suited to periodic systems, and have a long history of use in classical density functional theory calculations.”

In the paper

“However, this limitation is compensated by the ease of integral evaluation. Furthermore, the disadvantages of GTOs are less prominent in molecular calculations (Helgaker et al. , 2014).”

What the sources say

• Not checked yetcites Helgaker et al. , 2014

In the paper

“A similar reduction in the number of Hamiltonian terms can be obtained using gausslet basis sets (White, 2017), or 'discontinuous Galerkin' sets (McClean et al. , 2019a), which can both require fewer functions to accurately describe individual molecules than plane wave basis sets.”

What the sources say

• Not checked yetcites White, 2017
• Not checked yetcites McClean et al. , 2019a

In the paper

“Calculations using minimal basis sets are of limited accuracy, giving only a qualitative description of the system.”

In the paper

“Those authors also provide a technique to improve the energy estimate from a calculation by optimising the active space using orbital rotations, as is done in MCSCF calculations (Sec. III.C.2).”

What the sources say

• Not checked yetcites Takeshita et al. (2019)

In the paper

“The BK-tree mapping is uniquely defined even when the number of orbitals, M , is not a power of 2. As a result, when using the BK-tree mapping we are always able to use the qubit reduction by symmetry technique, which we will discuss in Sec. IV.C. We have observed that it is only possible to use this technique with the standard BK mapping when the number of orbitals is a power of two.”

In the paper

“Because of its product parametrization, the CC method generates a trial wavefunction which includes all possible determinants, albeit with an incorrect parametrization. It therefore provides faster convergence than the configuration interaction method.”

In the paper

“From the definition above, we can see that the diagonal elements of ρ 1 are the expectation values of the number operator for the corresponding orbitals.”

In the paper

“More accurate basis sets can be formed by adding increased radial flexibility to the valence orbitals (the orbitals of the highest occupied energy level), by considering a double-zeta representation of the valence orbitals. This can be achieved using split-valence (Ditchfield et al. , 1971) basis sets, such as the 6-31G basis.”

What the sources say

• Not checked yetcites Ditchfield et al. , 1971

In the paper

“As discussed in Sec. III.B.2, the full configuration interaction wavefunction contains a number of determinants which scales exponentially with the number of electrons, as roughly O ( M N ). As such, it requires an amount of memory that scales exponentially with the system size.”

In the paper

“STOn G basis sets are often called minimal basis sets, as they contain only the orbitals required to write the Hartree-Fock (HF) state (and those orbitals of similar energy).”

What the sources say

• Not checked yetcites Hehre et al. , 1969

In the paper

“This was improved by Berry et al. (2018), who used a circuit with O ( N log c 2 ( N )log 2 ( M )) gates, with a depth of O (log c 2 ( N )log 2 log 2 ( M )), where c ⩾ 1 and depends on the choice of sorting network used, and O ( N log 2 ( N )) ancilla qubits.”

What the sources say

• Not checked yetcites Berry et al. (2018)

In the paper

“The product parametrization also ensures size extensivity and size consistency.”

In the paper

“There are up to O ( N 2 M 4 ) possible 2-body terms, each leading to up to O (2 2log 2 ( M ) ) Pauli terms - meaning the Hamiltonian can contain up to O ( N 2 M 6 ) Pauli terms, which are up to 2log 2 ( M )-local.”

In the paper

“The number of qubits required to store the wavefunction using the different encoding methods. M is the number of spin-orbitals used in a basis set simulation, N is the number of simulated particles in the problem, and m = log 2 ( P ), where P is the number of grid points per axis for a grid based simulation.”

• Also stated in: § IV. QUANTUM COMPUTATIONAL CHEMISTRY MAPPINGS · p. 20 · plain

  “| First quantised, basis sets | N ⌈ log 2 ( M ) ⌉ |”

• Also stated in: § 1. Grid based methods · p. 20 · plain

  “As a result, it only requires (3 m + 1) N qubits to store the N electron wavefunction described by Eq. (40).”

In the paper

“A thorough comparison of the BK and JW mappings was performed by Tranter et al. (2018) for 86 molecular systems. They found that the BK transform was at least as efficient, in general, as the JW transform when finding the ground states of the molecular systems. In many cases using the BK transform made the calculations considerably more efficient.”

What the sources say

• Not checked yetcites Tranter et al. (2018)

In the paper

“While these basis sets result in an accurate description of the system with relatively few basis functions, they also lead to Hamiltonians containing up to O ( M 4 ) terms.”

In the paper

“The most common basis sets construct approximate STOs from linear combinations of GTOs. These approximate STOs are used as the basis functions for our atomic orbitals.”

In the paper

“The core orbitals have a single-zeta representation, while the valence orbitals have an n -zeta representation. The virtual orbitals considered are polarisation functions, with higher angular momenta than the valence orbitals. The polarisation functions are selected by the size of their contribution to the correlation energy.”

In the paper

“The original approach, by Abrams and Lloyd (1997), accomplished this using O ( N 2 log 2 2 ( M )) gates and O ( N log 2 ( M )) ancilla qubits.”

What the sources say

• Not checked yetcites Abrams and Lloyd (1997)

In the paper

“Because of the dependence on r 2 in the exponent, GTOs are more localised than STOs. As a result, GTOs do not approximate the atomic charge distribution as well as STOs, so more are required to describe a given orbital.”

In the paper

“However, those authors noted that such pathological cases can also exist for basis set methods, but are typically dealt with efficiently using a clever choice of basis function.”

What the sources say

• Not checked yetcites Kivlichan et al. (2017)

In the paper

“It is difficult to know the size of the gap along the entire adiabatic path a priori , restricts our ability to perform ASP in the minimum amount of time.”

In the paper

“Verstraete and Cirac (2005) developed a scheme to eliminate the strings of Z operators introduced by the JW transform, resulting in qubit operators with the same locality as the fermionic operators. This is achieved by doubling the number of qubits.”

What the sources say

• Not checked yetcites Verstraete and Cirac (2005)

In the paper

“Firstly, it is necessary for the register to initially be in a state with a non-zero overlap with the target eigenstate.”

In the paper

“There are also errors arising from constructing arbitrary gates from a discrete set of gates - such as approximating single qubit rotations from multiple T and Hadamard gates. These are typically referred to as circuit synthesis errors, ϵ CS , and can be quantified using techniques such as the Solovay-Kitaev theorem (Dawson and Nielsen, 2005).”

What the sources say

• Not checked yetcites Dawson and Nielsen, 2005

In the paper

“We note that most of the work to date in quantum computational chemistry has focused on second quantised basis set methods.”

In the paper

“Motta et al. (2018) utilised a low rank decomposition of the Hamiltonian in a Gaussian orbital basis to reduce the cost of time evolution. This decomposition is made possible by the structure present in the molecular Hamiltonian, which arises from the pairwise nature of the electron-electron interactions. Those authors leveraged a doubly factorised Cholesky decomposition of the Hamiltonian, with”

What the sources say

• Not checked yetcites Motta et al. (2018)

In the paper

“To realise the standard phase estimation algorithm given above, we sequentially need to time evolve the main register under the Hamiltonian H for times t 0 = 2 π, t 1 = 4 π, ..., t ω -1 = 2 ω π . The total coherent time evolution, T , is then given by approximately T = 2 ω +1 π .”

In the paper

“We remark that while this transformation leaves the ground state of the system unchanged, it does alter the excited states that can be found. In particular, we are restricted to finding those states with an electron number and total s z value equal to the values determined as described above.”

In the paper

“While first quantised basis set simulations require asymptotically fewer qubits than second quantised basis set simulations, for the smallest simulable systems (such as small molecules in minimal basis sets), second quantised basis set methods require either fewer qubits and/or shorter gate sequences.”

In the paper

“In the JW, parity and BK encoding methods, the number of qubits is equal to the number of spin-orbitals considered, M .”

In the paper

“Assuming fairly local interactions for a molecule, the degree of the graph will be less than the number of vertices. As a result, the BKSF transform may require fewer gates than the JW mapping.”

In the paper

“Using Eq. (61), for a success probability of p = 0 . 5, we require ω = n + 2 ancilla qubits. The total evolution time can be related to the binary precision ϵ PE = 1 / 2 n , to show that T = 8 π/ϵ PE . Given that our success probability for this estimate is p = 0 . 5, we expect to have to repeat the procedure twice to obtain a good estimate of the ground state. This is equivalent to a total of 16”

What the sources say

• Not checked yetcites Reiher et al. , 2017

In the paper

“As discussed above, both the canonical phase estimation algorithm and adiabatic state preparation require implementation of the time evolution operator, e -iHt , where H may or may not be time dependent.”

In the paper

“Reiher et al. (2017) noted that the true scaling may be closer to O ( M 2 / min t ∆( t )).”

What the sources say

• Not checked yetcites Reiher et al., 2017

In the paper

“It is important to note that the gate counts of all product formula based methods scale as O (poly(1 /ϵ )).”

In the paper

“Secondly, we must have a way to coherently implement a unitary operator defined by an efficiently invertible function of the Hamiltonian.”

In the paper

“The Taylor series and quantum signal processing + qubitization algorithms scale exponentially better with regards to the accuracy of the simulation than product formula based methods.”

In the paper

“For the case of a molecular Hamiltonian Reiher et al. (2017) showed that a number of calls scaling as π/ 2 ϵ PE will suffice.”

What the sources say

• Not checked yetcites Reiher et al. (2017)

In the paper

“The UCC ansatz is typically truncated at a given excitation level - usually single and double excitations (known as UCCSD).”

In the paper

“Firstly, the VQE is a heuristic method; it is currently unclear whether the quantum circuits proposed will yield better approximations of the ground state than the classical methods already available.”

In the paper

“Secondly, the number of measurements required to obtain the energy to the desired accuracy can be large.”

In the paper

“The UCC method retains all of the advantages of the CC method, with the added benefits of being fully variational, and able to converge when used with multireference initial states.”

In the paper

“McClean et al. (2018) showed that using hardware efficient ans¨ atze with random initial parameters makes the energy gradient essentially zero among most directions of Hilbert space, making classical optimisation extremely difficult.”

What the sources say

• Not checked yetcites McClean et al., 2018

In the paper

“The plane wave and plane wave dual basis sets reduce the number of terms in the Hamiltonian to O ( M 3 ) and O ( M 2 ), respectively.”

In the paper

“The VQE aims to find the lowest eigenvalue of a given Hamiltonian, such as that of a chemical system.”

In the paper

“As described in Sec. III.B.2, the Fermi-Hubbard Hamiltonian contains O ( M ) terms, many of which commute with each other.”

In the paper

“The VQE is a hybrid quantum-classical algorithm.”

In the paper

“The VQE relies upon the Rayleigh-Ritz variational principle.”

In the paper

“This has been improved to O ( M 3 N ) gates, with a depth of O ( M 2 N ) using swap networks (O'Gorman et al. , 2019)”

What the sources say

• Not checked yetcites O'Gorman et al., 2019

In the paper

“The UCC method is intractable on classical computers, but can be efficiently implemented on a quantum computer.”

What the sources say

• Not checked yetcites Yung et al. (2014)
• Not checked yetcites Peruzzo et al. (2014)

In the paper

“It appears to be robust against some errors (McClean et al. , 2016; O'Malley et al. , 2016), and capable of finding the ground state energies of small chemical systems using low depth circuits (Kandala et al. , 2017).”

What the sources say

• Not checked yetcites McClean et al., 2016
• Not checked yetcites O'Malley et al., 2016
• Not checked yetcites Kandala et al., 2017

In the paper

“Using the Taylor series method, algorithms were developed which scale as O ( M 5 ) (Babbush et al. , 2016) and O ( N 2 M 3 ) (Babbush et al. , 2017) for molecules in Gaussian basis sets.”

What the sources say

• Not checked yetcites Babbush et al., 2016
• Not checked yetcites Babbush et al., 2017

In the paper

“Direct search methods (such as Nelder-Mead simplex (Hempel et al. , 2018; Peruzzo et al. , 2014; Shen et al. , 2017), simulated annealing (Hempel et al. , 2018), particle swarm (Colless et al. , 2018; Santagati et al. , 2018) covariance matrix adaptation (Sagastizabal et al. , 2019), and the dividing rectangles algorithm (Kokail et al. , 2019)) were found to exhibit good convergence to the ground”

What the sources say

• Not checked yetcites Hempel et al., 2018
• Not checked yetcites Peruzzo et al., 2014
• Not checked yetcites Shen et al., 2017

In the paper

“McClean et al. (2016) showed that it is important to consider the covariances between Hamiltonian terms within groups, as if these covariances are positive, then it may be better to measure the terms separately.”

What the sources say

• Not checked yetcites McClean et al. (2016)

In the paper

“They reduced the number of gates required to implement a Trotter step of the Hamiltonian to O ( M 2 log 2 ( M )) with increasing molecular size, and O ( M 3 ) for fixed molecular size and increasing basis size.”

What the sources say

• Not checked yetcites Motta et al., 2018

In the paper

“Recent numerical work has explored the possibility of using classical neural networks to optimise VQE circuits for small instances of the Fermi-Hubbard model (Verdon et al. , 2019; Wilson et al. , 2019). This approach (known as 'meta-learning') works by training the neural networks on many random instances of the system being studied. The networks were able to successfully 'learn' to optimise the”

What the sources say

• Not checked yetcites Verdon et al., 2019
• Not checked yetcites Wilson et al., 2019

In the paper

“This leads to a scaling of O ( M 6 /ϵ 2 ) in a Gaussian orbital basis, and O ( M 4 /ϵ 2 ) for a plane wave dual basis (Babbush et al. , 2018c; Cao et al. , 2019; McClean et al. , 2014).”

What the sources say

• Not checked yetcites Babbush et al., 2018c
• Not checked yetcites Cao et al., 2019
• Not checked yetcites McClean et al., 2014

In the paper

“Moreover, we can use classically tractable methods to get initial approximations for the remaining non-zero parameters (O'Malley et al. , 2016; Romero et al. , 2019), which makes the classical optimisation step of the VQE easier.”

What the sources say

• Not checked yetcites O'Malley et al., 2016
• Not checked yetcites Romero et al., 2019

In the paper

“The simultaneous perturbation stochastic approximation (SPSA) algorithm, has also been used to successfully find the ground state energy of small molecules, despite relatively large uncertainties due to shot noise, and physical error rates (Ganzhorn et al. , 2018; Kandala et al. , 2017, 2018).”

What the sources say

• Not checked yetcites Ganzhorn et al., 2018
• Not checked yetcites Kandala et al., 2017
• Not checked yetcites Kandala et al., 2018

In the paper

“It has also been noted (Kivlichan et al. , 2018) that it is possible to use locality preserving mappings (Jiang et al. , 2018a; Verstraete and Cirac, 2005) to perform Trotter steps in constant depth. However, this comes at a cost of requiring additional qubits, and may have a large constant factor gate overhead.”

What the sources say

• Not checked yetcites Kivlichan et al., 2018
• Not checked yetcites Jiang et al., 2018a
• Not checked yetcites Verstraete and Cirac, 2005

In the paper

“This problem was partly mitigated by using a good initialisation strategy, such as using a chemically motivated (M¨ ollerPlesset) guess for UCC excitation amplitudes (Romero et al. , 2019).”

What the sources say

• Not checked yetcites Romero et al., 2019

In the paper

“The approach also demonstrated some resilience to noise included in the simulations (Wilson et al. , 2019).”

What the sources say

• Not checked yetcites Wilson et al., 2019

In the paper

“Direct search algorithms are considered more robust to noise than gradient based methods, but may require more function evaluations (Kolda et al. , 2006).”

What the sources say

• Not checked yetcites Kolda et al., 2006

In the paper

“Kivlichan et al. (2018) showed that for Hamiltonians of this form, we can implement Trotter steps in depth O ( M ), using O ( M 2 ) two qubit gates.”

What the sources say

• Not checked yetcites Kivlichan et al. (2018)

In the paper

“Jiang et al. (2018b) improved this result for the case of a 2D array of qubits with nearest-neighbour connectivity. They showed that it is possible to prepare initial states of the Fermi-Hubbard model using O ( M 1 . 5 ) gates, and perform Trotter steps of the Hamiltonian using O ( M ) gates for each Trotter step.”

What the sources say

• Not checked yetcites Jiang et al. (2018b)

In the paper

“A limitation of the majority of these studies is that realistic noise has rarely been considered, so it is still unclear exactly how effective each method may be in practice.”

In the paper

“However, this approach has been questioned by Grimsley et al. (2019) who found that depending on how the operators in the Trotterized UCCSD ansatz are ordered, the optimised energies of systems displaying significant electron correlation can vary by amounts larger than chemical accuracy.”

What the sources say

• Not checked yetcites Grimsley et al. (2019)

In the paper

“Some authors have also considered using additional unitary transforms to enable more terms to be grouped, reducing the number of groups of terms from O ( M 4 ) to O ( M 3 ) (Crawford et al. , 2019; Gokhale et al. , 2019; Gokhale and Chong, 2019; Jena et al. , 2019; Yen et al. , 2019b; Zhao et al. , 2019).”

What the sources say

• Not checked yetcites Crawford et al., 2019
• Not checked yetcites Gokhale et al., 2019
• Not checked yetcites Gokhale and Chong, 2019

In the paper

“This technique can be generalised to any ansatz with a repeating, layered structure.”

In the paper

“The extrapolation method (Li and Benjamin, 2017; Temme et al. , 2017) works by intentionally increasing the dominant error rate, ϵ 0 , by a factor λ , and inferring the error free result by extrapolation.”

What the sources say

• Not checked yetcites Li and Benjamin, 2017
• Not checked yetcites Temme et al., 2017

In the paper

“It is important to note that error mitigation schemes are not a scalable solution to the problem of noise in quantum hardware. In order to scale up computations to arbitrarily large sizes, fault-tolerant, error corrected quantum computers are required.”

In the paper

“Overall, simulated VQE calculations on small molecules were sped up by a factor of 2 to 4 (Barkoutsos et al. , 2018).”

What the sources say

• Not checked yetcites Barkoutsos et al., 2018

In the paper

“These contraction VQE methods differ from the overlap based methods, in that they obtain all of the eigenstates at the same time, rather than sequentially. Consequently, all eigenstates should be obtained to a similar accuracy.”

In the paper

“The SWAP test approach requires a circuit that has twice as many gates as the ansatz used (but is of the same depth), and has twice as many qubits.”

In the paper

“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).”

What the sources say

• Not checked yetcites Huo and Li, 2018

In the paper

“Because only the correlation energy is calculated, fewer measurements are required and classical optimisation becomes easier.”

In the paper

“These techniques are only effective when used with low depth circuits, such that the total error rate in the circuit is low.”

In the paper

“The quantum subspace expansion (QSE) method was originally introduced to find excited states (McClean et al. , 2017a), but has proven to be one of the most useful techniques introduced to near-term quantum computational chemistry, with benefits including better ground state estimation (discussed below) and mitigation of hardware errors (discussed in Sec. VI.C).”

What the sources say

• Not checked yetcites McClean et al., 2017a

In the paper

“The quantum subspace expansion (QSE) (McClean et al. , 2017a) described in Sec. V.C.1 can mitigate errors in the VQE, in addition to calculating the excited energy eigenstates.”

What the sources say

• Not checked yetcites McClean et al. , 2017a

In the paper

“This method is most effective at correcting systematic errors, but can also suppress some stochastic errors.”

In the paper

“The probabilistic error cancellation method has been experimentally demonstrated on both superconducting (Song et al. , 2018) and trapped ion (Zhang et al. , 2019) systems.”

What the sources say

• Not checked yetcites Song et al. , 2018
• Not checked yetcites Zhang et al. , 2019

In the paper

“The overhead γ determines the number of additional measurements required to keep the variance of the error mitigated result equal to the variance of the noisy result.”

In the paper

“They found the EoM method to be robust to noise when demonstrated experimentally, and more accurate for finding excitation energies than the QSE, in numerical tests.”

What the sources say

• Not checked yetcites Ollitrault et al. (2019)