Abstract
The quantum simulation of quantum chemistry is a promising application of quantum computers. However, for N molecular orbitals, the \({\mathcal{O}}({N}^{4})\) gate complexity of performing Hamiltonian and unitary Coupled Cluster Trotter steps makes simulation based on such primitives challenging. We substantially reduce the gate complexity of such primitives through a twostep lowrank factorization of the Hamiltonian and cluster operator, accompanied by truncation of small terms. Using truncations that incur errors below chemical accuracy allow one to perform Trotter steps of the arbitrary basis electronic structure Hamiltonian with \({\mathcal{O}}({N}^{3})\) gate complexity in small simulations, which reduces to \({\mathcal{O}}({N}^{2})\) gate complexity in the asymptotic regime; and unitary Coupled Cluster Trotter steps with \({\mathcal{O}}({N}^{3})\) gate complexity as a function of increasing basis size for a given molecule. In the case of the Hamiltonian Trotter step, these circuits have \({\mathcal{O}}({N}^{2})\) depth on a linearly connected array, an improvement over the \({\mathcal{O}}({N}^{3})\) scaling assuming no truncation. As a practical example, we show that a chemically accurate Hamiltonian Trotter step for a 50 qubit molecular simulation can be carried out in the molecular orbital basis with as few as 4000 layers of parallel nearestneighbor twoqubit gates, consisting of fewer than 10^{5} nonClifford rotations. We also apply our algorithm to iron–sulfur clusters relevant for elucidating the mode of action of metalloenzymes.
Similar content being viewed by others
Introduction
The electronic structure (ES) problem, namely, solving for the ground or lowlying eigenstates of the Schrödinger equation for atoms, molecules, and materials, is an important problem in theoretical chemistry and physics. There are several approaches to solving this problem on a quantum computer, including projecting approximate solutions to eigenstates using phase estimation^{1,2,3}, directly preparing eigenstates using the adiabatic algorithm^{4,5,6}, or using quantum variational algorithms^{7,8} to optimize parameterized circuits corresponding to unitary Coupled Cluster (uCC)^{9,10,11} or approximate adiabatic state preparation^{12,13}.
Time evolution, under the Hamiltonian or the uCC cluster operator, is a common component in these algorithms. For nearterm quantum devices (especially with limited connectivity), TrotterSuzuki based methods for time evolution, (which break the evolution of a complex sum of operators such as the Hamiltonian into a sequence of Trotter steps each evolving only a single term in the sum) are most compelling since they lack the complex controlled operations required by asymptotically more precise methods^{14,15,16}. In order to perform a discrete simulation, the Hamiltonian or cluster operator is first represented in a singleparticle basis of dimension N. However, in many bases, including the molecular orbital and active spaces bases common in ES, the Hamiltonian and cluster operator contain \({\mathcal{O}}({N}^{4})\) secondquantized terms. This leads to at least \({\mathcal{O}}({N}^{4})\) gate complexity for a single Trotter step^{17,18}, a formidable barrier to practical progress. While complexity can be reduced using alternative bases^{19,20}, such representations are not usually as compact as the molecular orbital one (i.e. a longer basis expansion is required to represent typical physical states of interest). Thus, reducing the cost of the Trotter step for general bases is an important goal, particularly within the context of nearterm simulation paradigms.
In this article, we introduce a method to rewrite the Trotter step of a general quantum chemistry Hamiltonian evolution, as well other exponentials of quartic fermionic operators such as the uCC operator, solely in terms of unitary singleparticle rotations and Trotter steps of twobody operators. This allows for the efficient implementation of Trotter steps on a linearly connected quantum device within the JordanWigner fermionic encoding^{20,21}. It further allows for a systematic lowrank truncation (whose applicability derives from sparsity in the number of terms in the sum) which can executed in a gatecount efficient manner by the above class of circuits. The procedure starts from a nested matrix factorization of the fourbody twoelectron interaction term, related to the one described in the Hamiltonian evolution context by Poulin et al.^{22}. A key additional idea is that the nested matrix factorization exposes a lowrank structure when the interaction term is a physical operator. This was observed empirically in the classical electronic structure context for the Hamiltonian by Peng and Kowalski in ref. ^{23} (although that work obtained an incorrect empirical scaling) and studied more deeply by some of us in ref. ^{24}, where we presented mathematical evidence for the correct scaling. Here, we demonstrate that the lowrank structure allows one to perform truncations that significantly reduce the gate complexity of the Trotter step for the Hamiltonian operator, as well as for the unitary cluster operator. In particular, we achieve a Hamiltonian Trotter step with an asymptotic gate complexity scaling as \({\mathcal{O}}({N}^{2})\) with system size, and \({\mathcal{O}}({N}^{3})\) for fixed systems and increasing basis size. These scalings require only linear nearestneighbor connectivity. We give numerical evidence that we can carry out a Hamiltonian Trotter step on a 50 qubit quantum chemical problem with as few as 4000 layers of twoqubit gates on a linear nearestneighbor architecture, a viable target for implementation on nearterm quantum devices. Compiled to Clifford gates and singlequbit rotations, this requires fewer than 10^{5} nonClifford rotations, an improvement of orders of magnitude over past Trotterbased methods in a faulttolerant cost model^{25}.
Results
Doubledecomposition structure
We first define the Hamiltonian H and cluster operator τ. In second quantization H is
where \({a}_{p}^{\dagger }\) and a_{p} are fermionic creation and annihilation operators for spin orbital ϕ_{p}, and the scalar coefficients h_{pq} and h_{pqrs} are the one and twoelectron integrals over the basis functions ϕ_{p} (here assumed real).
The uCC cluster operator τ = T − T^{†}, where T is the standard (nonunitary) coupled cluster (CC) operator. For uCCSD (uCC with single and double excitations applied to a single determinant reference),
where ij, ab index the N_{o} occupied and N_{v} virtual spin orbitals respectively, and N_{o} + N_{v} = N. For the scaling arguments with system size, we assume N_{o}, N_{v} ∝ N, while increasing basis size corresponds to increasing N_{v} only. Both H and τ contain \({\mathcal{O}}({N}^{4})\) secondquantized terms. Thus, for arbitrary Hamiltonian integrals or cluster amplitudes, regardless of the gate decomposition or fermion encoding used, implementing the timeevolution Trotter step requires at least \({\mathcal{O}}({N}^{4})\) gates.
The integrals and cluster amplitudes that one encounters in molecular ES applications, however, are not arbitrary, but contain considerable structure. We now show that this allows us to construct approximate operators \(H^{\prime}\) or \(\tau ^{\prime}\), accurate to within a desired tolerance ε, that can be implemented with greatly reduced gate counts. The physical basis for this result is the pairwisenature of the Hamiltonian interactions, arising from the 1/r_{12} Coulomb kernel in realspace. More precisely, we will rewrite the twofermion parts of H and iτ associated with the integrals h_{pqrs} and \({t}_{pqrs}^{\prime}\) as a doublefactorized form
where, defining \({\psi }_{i}^{(\ell )}=\mathop{\sum }\nolimits_{p = 1}^{N}{U}_{pi}^{(\ell )}{\phi }_{p}\),
are number operators in a rotated basis. Approximate \(H^{\prime}\) and \(\tau ^{\prime}\) with reduced complexity can then be obtained by truncating the summations over L, ρ_{ℓ}. The dependence of the error ε on L and ρ_{ℓ} is discussed further below.
The doublydecomposed form of V can be obtained using a nested matrix factorization, a type of tensor factorization introduced in ref. ^{23}. We illustrate this for the Hamiltonian operator. First, the creation and annihilation operators are reordered,
and \(V^{\prime}\) is recast into a supermatrix indexed by orbitals (ps), (qr) involving electrons 1,2 respectively. Due to the eightfold symmetry h_{pqrs} = h_{srqp} = h_{pqsr} = h_{qprs} = h_{qpsr} = h_{rsqp} = h_{rspq} = h_{srpq} this matrix is real symmetric, thus we can write a matrix decomposition in terms of a rankthree auxiliary tensor \({\mathcal{L}}\) such that
A simple way to obtain \({\mathcal{L}}\) is to diagonalize the \(V^{\prime}\) supermatrix, although other techniques^{26,27,28,29,30,31,32}, such as Cholesky decomposition (CD), are also commonly used; we use the Cholesky decomposition in our numerical simulations below. (Note that the positive nature of the Coulomb kernel means that \(V^{\prime}\) is positive also, as is used in the Cholesky decomposition). The next step is to decompose each auxiliary matrix \({{\mathcal{L}}}^{(\ell )}\). Again for the Hamiltonian, this is also real symmetric, thus we can similarly diagonalize it,
where λ^{(ℓ)}, U^{(ℓ)} are the eigenvalues and eigenvectors of \({{\mathcal{L}}}^{(\ell )}\). Combining the two eigenvalue decompositions yields the doublefactorized result, Eq. (3).
In the cluster operator, amplitudes t have fourfold mixed symmetry and antisymmetry, t_{abij} = t_{jiba} = −t_{baij} = − t_{abji}. Thus because the cluster operator is not simply symmetric, we cannot use a Cholesky decomposition and must modify the arguments above. However, as shown in the Supplementary Discussion, using a singular value decomposition of the cluster operator, we can write \(i\tau ={\sum }_{\ell \mu }{Y}_{\ell ,\mu }^{2}\), where Y_{ℓ,μ} are normal and can be diagonalized giving the same doublefactorized form.
Accuracy of lowrank approximations
For an exact decomposition of H, L = N^{2} and ρ_{ℓ} = N. However, it is well established from empirical ES calculations that the ranks L and ρ_{ℓ} can be significantly reduced if we approximate H by truncating small terms. The lowrank truncations are performed as follows: in the case of L, we truncate the CD in the AO basis or localized MO basis based on the L^{∞} norm, i.e. use the smallest L such that \({\max }_{psqr} {h}_{ps,qr}\mathop{\sum }\nolimits_{\ell = 1}^{L}{{\mathcal{L}}}_{ps}^{(\ell )}{{\mathcal{L}}}_{qr}^{(\ell )} \,<\,{\varepsilon }_{\text{CD}}\). The computational cost of the modified Cholesky decomposition scheme is known to scale asymptotically like \({\mathcal{O}}({N}^{3})\) within the AO basis for a fixed error threshold^{33}. For ρ_{ℓ}, we perform an eigenvalue truncation (ET) based on the L^{1} norm, i.e. use the smallest ρ_{ℓ} such that \(\mathop{\sum }\nolimits_{j = {\rho }_{\ell }+1}^{N} {\lambda }_{j}^{(\ell )}\, <\,{\varepsilon }_{\text{ET}}\). Note that for H, ε_{CD} and ε_{ET} have dimension energy and square root of energy. For simplicity, we have chosen ε_{CD} = ε_{ET} ≡ ε in atomic units. For this truncation of H it has been shown that when increasing the molecular size or simulation basis, \(L \sim {\mathcal{O}}(N)\), while \(\langle {\rho }_{\ell }\rangle =\frac{1}{L}\mathop{\sum }\nolimits_{\ell = 1}^{L}{\rho }_{\ell } \sim {\mathcal{O}}(1)\) for increasing molecular size in the asymptotic limit^{24}.
For the uCC operator, the amplitudes t can be cast in a supermatrix t_{ai,bj} that is symmetric, t_{ai,bj} = t_{bj,ai}, but not positive. Therefore, we substitute the Cholesky decomposition with a singular value decomposition, obtaining \(L \sim {\mathcal{O}}(N)\) with increasing basis size but \(L \sim {\mathcal{O}}({N}^{2})\) with increasing molecular size (albeit with a small coefficient); the scaling properties of ρ_{ℓ,μ} have not previously been studied. Note that, for the uCC operator, the truncations ε_{SVD}, ε_{ET} are dimensionless, unlike for H.
In Fig. 1 we show L and 〈ρ_{ℓ}〉 for different truncation thresholds in: (set 1) a variety of molecules that can be represented with a modest number of qubits (CH_{4}, H_{2}O, CO_{2}, NH_{3}, H_{2}CO, H_{2}S, F_{2}, BeH_{2}, HCl) using STO6G, ccpVDZ, 631G*, ccpVTZ bases; (set 2) alkane chains C_{n}H_{2n+2}, n ≤ 8, using the STO6G basis; (set 3) Fe–S clusters ([2Fe–2S], [4Fe–4S], and the nitrogenase P^{N} cluster, in active spaces with N = 40, 72, 146 respectively. Further details the of calculations are given in the Methods section.
We first give some context to these sets for quantum simulation. The valence active space models considered in the Fe–S systems in set 3 are representative of a nearerterm quantum application where not all degrees of freedom are treated on a quantum computer. In this set, the underlying Gaussian basis dependence is largely removed by the reduction to an active space, as such calculations converge exponentially quickly with basis size^{34}. All electron simulations of molecules of the kind in set 1 and 2 may be considered in the context of quantum resources available in the longerterm. While we have considered only a representative set of systems, additional intuition for the Hamiltonian ranks in these classes of molecules can be obtained from quantum Monte Carlo calculations, which work well for sets 1 and 2, and where a similar decomposition has been applied^{24}.
For the uCC operator, in order to treat sufficiently large systems to observe the scaling trends, we have used the (classically computable) traditional CC amplitudes, equal to the uCC amplitudes in the weakcoupling limit (they agree through third order in perturbation theory). The uCC and traditional CC amplitudes are thus similar for all molecules in sets 1 and 2 near their equilibrium geometries, and molecules in set 3 in the highest spin electronic state.
For Hamiltonian evolution, we clearly see the L ∝ N scaling across different truncation thresholds, for both increasing system size and basis. For τ, L ∝ N with increasing basis in a fixed molecule, while L ∝ N^{2} with increasing size (e.g. in alkane chains). Interestingly, the value of L in the Hamiltonian decomposition is quite similar across different molecules for the same number of spin orbitals (qubits). In the subsequent ET for the Hamiltonian, 〈ρ_{ℓ}〉 features sublinear scaling for set 2 (alkanes, n ≥ 5, represented here with 75–125 qubits), as well as for set 3 (Fe–S clusters). While we have shown in 1D systems that ρ_{ℓ} ~ O(1) rigorously in the asymptotic size regime, these systems are still too small to see this saturation, although the practical reduction in ρ_{ℓ} from full rank in sets 2 and 3 is significant. For the uCC operator, we observe that 〈ρ_{ℓ}〉 scales like \({\mathcal{O}}(N)\) for alkane chains and increasing molecular size, while it is approximately constant for increasing basis set size. The less favorable scaling of L, 〈ρ_{ℓ,μ}〉 with system size for the uCC operator, relative to H, stems from the antisymmetry properties of the amplitudes, which in the current factorization means that Y_{ℓ,μ} do not show the same sparsity as \({{\mathcal{L}}}^{(\ell )}\).
Firstorder correction
The error arising from the truncations leading to \(H^{\prime}\) and \(\tau ^{\prime}\) can be understood in terms of two components: (i) the error in the operators, and (ii) the error in the states generated by time evolution with these operators. It is possible to substantially reduce both errors using quantities that can be computed classically. We illustrate this for the error in \(H^{\prime}\). First, the correlation energy, defined as E_{c} = E − E_{HF} with E the total energy and E_{HF} the Hartree–Fock energy, is usually a much smaller quantity than the total energy in chemical systems. It is expected to be subject to much smaller truncation errors, mainly due to cancellation of errors between exact and meanfield truncations. Thus, using the classically computed meanfield energy of \(H^{\prime}\), we can obtain the truncated correlation energy as \({E}_{\text{c}}^{\prime} =E^{\prime} {E}_{\text{HF}}^{\prime}\). Second, we can estimate the remaining error in \({E}_{c}^{\prime}\) from firstorder perturbation theory as \(\langle \psi  HH^{\prime}  \psi \rangle\), where a classical approximation to ψ is used. If the classical ψ is accurate, the corrected \({E}_{c}^{\prime}\) is then accurate to \({\mathcal{O}}({\varepsilon }^{2})\). In Fig. 2 we plot \( {E}_{\,\text{c}}^{\prime}{E}_{\text{c}}\) for H_{2}O at the ccpVDZ level. Adding the perturbative correction from the classical CC groundstate reduces the error by about an order of magnitude, such that even an aggressive truncation threshold of ε_{CD} = ε_{TH} ≡ ε = 10^{−2} a.u. yields the total correlation energy within the standard chemical accuracy of 1.6 × 10^{−3} a.u.
For the \({\tau }^{\prime}\) truncation, one could include a similar error correction for the correlation energy derived from approximate cluster amplitudes (although we do not do so here). Note that we have only considered taking a given amplitude \(\tau ^{\prime}\) and the error from implementing the corresponding uCC operator with truncation. However, there is the additional possibility of optimizing the amplitude \({\tau }^{\prime}\) within the truncated form in a variational uCC approach. In this case, the stationary condition can formally be obtained by differentiating through the ansatz. While we reserve a detailed error analysis in this setting for future work, so long as one is close to the variational minimum, it is clear that the resulting error in the variational energy remains quadratic with respect to small truncations of \({\tau }^{\prime}\).
Gate counts for quantum computers
The doublefactorized decomposition Eq. (3) provides a simple circuit implementation of the Trotter step. For example, for the Hamiltonian Trotter step, we write
where \({\tilde{U}}^{(\ell )}={U}^{\dagger (\ell )}{U}^{(\ell +1)}\). Time evolution then corresponds to (singleparticle) basis rotations with evolution under the singleparticle operator h + S and pairwise operators V^{(ℓ)}. Note that because h + S is a onebody operator, it can be exactly implemented (with Trotter approximation) using a singleparticle basis change U^{(0)} followed by a layer of N phase gates (the latter being a simultaneous application of onequbit gates to all qubits). The singleparticle basis changes U^{(ℓ)} can be implemented using \(\left(\begin{array}{c} {N} \\ {2}\end{array} \right)  \left(\begin{array}{c}{N  \rho_\ell}\\{2}\end{array}\right)\) Givens rotations^{35} (detailed in the Supplementary Discussion). These rotations can be implemented efficiently using twoqubit gates on a linearly connected architecture^{20,21}.
Taking into account S_{Z} spin symmetry to implement basis rotations separately for spinup and spindown orbitals gives a count of \(2\left(\begin{array}{c}N/2\\ 2\end{array}\right)  2\left(\begin{array}{c}(N{\rho }_{\ell })/2\\ 2\end{array}\right)\) with a corresponding circuit depth (on a linear architecture) of (N + ρ_{ℓ})/2. Using a fermionic swap network, a Trotter step corresponding to evolution under the pairwise operator V^{(ℓ)} can be implemented in \(\left(\begin{array}{c}{\rho }_{\ell }\\ 2\end{array}\right)\) linear nearestneighbor twoqubit gates, with a twoqubit gate depth of exactly ρ_{ℓ}.
Summing these terms, counts thus are \(\left(\begin{array}{c}N\\ 2\end{array}\right)+{\sum }_{\ell \mu }\left[\right.\left(\begin{array}{c}N\\ 2\end{array}\right)\left(\begin{array}{c}N{\rho }_{\ell ,\mu }\\ 2\end{array}\right)\left]\right.+\left(\begin{array}{c}{\rho }_{\ell ,\mu }\\ 2\end{array}\right)\), where the μ subscript can be ignored when considering the Hamiltonian.
To realize this algorithm on a nearterm device, where the critical cost model is the number of twoqubit gates, one can either implement the gates directly in hardware^{36}, which requires \(\mathop{\sum }\nolimits_{\ell = 1}^{L}\left[\right.\frac{N{\rho }_{\ell }}{4}+\frac{{\rho }_{\ell }^{2}}{4}{\rho }_{\ell }\left]\right.\) gates on a linear nearestneighbor architecture, with circuit depth \(\mathop{\sum }\nolimits_{\ell = 1}^{L}\frac{N}{2}+\frac{3{\rho }_{\ell }}{2}\). If decomposing into a standard twoqubit gate set (e.g. CZ or CNOT), the gate count would be three times the above count.
To realize this algorithm within an errorcorrected code such as the surface code^{37}, where the critical cost model is the number of T gates, one can decompose each Givens rotation gate in two arbitrary singlequbit rotations and each diagonal pair interaction in one arbitrary singlequbit rotation. Thus, the number of singlequbit rotations is \(\mathop{\sum }\nolimits_{\ell = 1}^{L}\left[\right.\frac{N{\rho }_{\ell }}{2}2{\rho }_{\ell }\left]\right.\). Using standard synthesis techniques for singlequbit rotations, the number of T gates depends on the desired precision as \(1.15{{\mathrm{log}}\,}_{2}(1/{\varepsilon }_{\text{RS}})+9.2\) times this count^{38}, where ε_{RS} is the tolerance of rotation synthesis. Note that while defining ε_{RS} is needed to obtain the final T gate count (see e.g. ref. ^{39}), if we assume a given synthesis threshold, the relative cost of two algorithms in the errorcorrected cost model is obtained simply by comparing the number of singlequbit rotations.
In Fig. 3 we show the total gate counts needed to carry out a Trotter step of \(H^{\prime}\) and \(\tau ^{\prime}\) with different truncation thresholds. Using the scaling estimates obtained above for L, ρ_{ℓ} in the gate count expression, we expect the Hamiltonian Trotter step to have a gate count \({N}_{\text{gates}} \sim {\mathcal{O}}({N}^{2})\) for increasing molecular size, and \({\mathcal{O}}({N}^{3})\) for fixed molecular size and increasing basis size, and the uCC Trotter step to show \({N}_{\text{gates}} \sim {\mathcal{O}}({N}^{4})\) for increasing molecular size and \({\mathcal{O}}({N}^{3})\) with increasing basis size. This scalings are confirmed by the gate counts in Fig. 3. As seen, the crossover between N^{3} and N^{2} behavior of the Hamiltonian Trotter gate cost, for alkanes (set 2), occur at larger N than one would expect from the 〈ρ_{ℓ}〉 data alone from Fig. 1, due to tails in the distribution of ρ_{ℓ}.
The threshold for classicalquantum crossover in recent quantum supremacy experiments (although dependent on the precise computational task) has been studied in detail at roughly 50 qubits, see for example ref. ^{40}. For nearterm devices, the number of layers of gates on a parallel architecture with restricted connectivity is often considered a good cost model. Using the circuit depth estimate ∑_{ℓ}(N + ρ_{ℓ}), we see that we can carry out a single Hamiltonian Trotter step on a system with 50 qubits with as few as 4000 layers of parallel gates on a linear architecture. Within cost models appropriate for error correction, the most relevant cost metric is the number of T gates^{25,41,42}. For our algorithms, T gates enter through singlequbit rotations and thus, the number of nonClifford singlequbit rotations is an important metric. Based on the gate count estimate for basis changes, the number of nonClifford rotations required for our Trotter steps is roughly 100,000. For a fixed ε_{RS} = 10^{−6}, the number of T gates obtained after rotation synthesis would then be approximately 30 times this number.
Discussion
In summary, we have introduced a nested decomposition of the Hamiltonian and uCC operators, leading to substantially reduced gate complexity for the Trotter step both in realistic molecular simulations with under 100 qubits, and in the asymptotic regime. The discussed decomposition is by no means the only one possible and, for the uCC operator, it is nonoptimal, as more efficient decompositions for antisymmetric quantities exist^{43}. Future work to better understand the interplay between classical tensor decompositions and the components of quantum algorithms thus presents an exciting possibility for further improvements in practical quantum simulation algorithms.
Note: since the time this paper was first posted as a preprint, many other works have further applied the decomposition in this work, or very closely related decompositions. Some examples of such applications include the implementation of a clusterJastrow ansatz^{44}; its use as a component in efficient Hamiltonian evolution in conjunction with other techniques and under different cost models^{45,46}; and use of this form to reduce the cost of measurements^{47,48}).
Methods
The Hartree–Fock calculations for the small molecules were obtained using chemistry package PySCF^{49}, and calculations for the iron–sulfur clusters were obtained using densitymatrix renormalization group (DMRG)^{50,51} implemented within PySCF as BLOCK.
Details of calculations
Here, we provide further details about the calculations yielding the data shown in the main text, focusing on each of the three studied sets (set 1, set 2, set 3).
Set 1—comprises 9 small molecules (namely CH_{4}, H_{2}O, CO_{2}, NH_{3}, H_{2}CO, H_{2}S, F_{2}, BeH_{2}, HCl), studied at experimental equilibrium geometries from ref. ^{52}. Molecules in this set have been studied with restricted Hartree–Fock (RHF) and restricted classical coupled cluster with single and double excitations (RCCSD) on top of the RHF state. Matrix elements of the Hamiltonian and classical RCCSD amplitudes have been computed with the PySCF software^{49}, using the STO6G, 631G*, ccpVDZ, ccpVTZ bases.
Set 2—comprises alkane chains (namely ethane, propane, butane, pentane, hexane, heptane and octane, all described by the chemical formula C_{n}H_{2n+2} with n = 2…8), studied at experimental equilibrium geometries from ref. ^{52}. Molecules in this set have been studied with RHF, RCCSD methods. Matrix elements of the Hamiltonian and classical RCCSD amplitudes have been computed with the PySCF software^{49}, using the STO6G basis.
Set 3—comprises Fe–S clusters [2Fe–2S] [2Fe(II)] and [4Fe–4S] [2Fe(III),2Fe(II)], and the P^{N}cluster [8Fe–7S] [8Fe(II)]) of nitrogenase.
The active orbitals of [2Fe–2S] and [4Fe–4S] complexes were prepared by a split localization of the converged molecular orbitals at the level of BP86/TZPDKH, while those of the [8Fe–7S] were prepared at the level of BP86/def2SVP. The active space for each complex was composed of Fe 3d and S 3p of the core part and σbonds with ligands, which is the minimal chemically meaningful active space. The structure of the iron–sulfur core and the numbers of active orbitals and electrons for each complex are summarized in Fig. 4.
Molecules in this set were treated with densitymatrix renormalization group (DMRG)^{50,51}, using the PySCF software. The DMRG calculations were performed for the S = 0 states, which are the experimentally identified ground states, with bond dimensions 8000, 4000, and 2000 for [2Fe–2S], [4Fe–4S], and [8Fe–7S]. Note that the active space employed in the present work for the P^{N}cluster is larger than the active space previously used to treat the FeMoco cluster of nitrogenase, having the same number of transition metal atoms^{25}.
Brokensymmetry unrestricted Hartree–Fock (UHF) (M_{S} = 0) calculations were carried out for [2Fe–2S] and [4Fe–4S]. For [8Fe–7S], due to convergence issues, highspin UHF calculations were used instead.
Data availability
Data regarding the electronic structure calculations can be provided upon request. The matrix elements h_{pq}, h_{pqrs} for the iron–sulfur cluster are made available in a compressed archive form (FeS_integrals.tar).
Code availability
Code performing the doubledecomposition and electronic structure calculations are available upon request. PySCF is available on GitHub (https://github.com/pyscf/pyscf).
References
Kitaev, A. Y. Quantum measurements and the Abelian stabilizer problem. Electron. Colloquium Comput. Complex. 3, Preprint at: http://arxiv.org/abs/quantph/9511026 (1996).
Abrams, D. S. & Lloyd, S. Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors. Phys. Rev. Lett. 83, 5162 (1999).
AspuruGuzik, A., Dutoi, A. D., Love, P. J. & HeadGordon, M. Simulated quantum computation of molecular energies. Science 309, 1704 (2005).
Farhi, E. et al. A quantum adiabatic evolution algorithm applied to random instances of an NPcomplete problem. Science 292, 472 (2001).
Wu, L.A., Byrd, M. & Lidar, D. Polynomialtime simulation of pairing models on a quantum computer. Phys. Rev. Lett. 89, 057904 (2002).
Babbush, R., Love, P. J. & AspuruGuzik, A. Adiabatic quantum simulation of quantum chemistry. Sci. Rep. 4, 6603 (2014).
Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5, 4213 (2014).
O’Malley, P. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016).
Kutzelnigg, W. Quantum chemistry in Fock space. I. The universal wave and energy operators. J. Chem. Phys. 77, 3081 (1982).
Yanai, T. & Chan, G. K.L. Canonical transformation theory for multireference problems. J. Chem. Phys. 124, 194106 (2006).
Romero, J. et al. Strategies for quantum computing molecular energies using the unitary coupled cluster ansatz. Quantum Sci. Technol. 4, 014008 (2018).
Farhi, E., Goldstone, J. & Gutmann, S. A quantum approximate optimization algorithm. Preprint at: https://arxiv.org/abs/1411.4028 (2014).
Wecker, D., Hastings, M. B. & Troyer, M. Progress towards practical quantum variational algorithms. Phys. Rev. A 92, 042303 (2015a).
Berry, D. W., Childs, A. M., Cleve, R., Kothari, R. & Somma, R. D. Simulating Hamiltonian dynamics with a truncated Taylor series. Phys. Rev. Lett. 114, 090502 (2015).
Low, G. H. & Chuang, I. L. Hamiltonian simulation by qubitization. Quantum 3, 163 (2019).
Low, G. H. & Wiebe, N. Hamiltonian simulation in the interaction picture. Preprint at: https://arxiv.org/abs/1805.00675 (2018).
Seeley, J. T., Richard, M. J. & Love, P. J. The BravyiKitaev transformation for quantum computation of electronic structure. J. Chem. Phys. 137, 224109 (2012).
Hastings, M. B., Wecker, D., Bauer, B. & Troyer, M. Improving quantum algorithms for quantum chemistry. Quantum Inf. Comput. 15, 1–21 (2015).
Babbush, R. et al. Lowdepth quantum simulation of materials. Phys. Rev. X 8, 011044 (2018a).
Kivlichan, I. D. et al. Quantum simulation of electronic structure with linear depth and connectivity. Phys. Rev. Lett. 120, 110501 (2018).
Jiang, Z., Sung, K. J., Kechedzhi, K., Smelyanskiy, V. N. & Boixo, S. Quantum algorithms to simulate manybody physics of correlated fermions. Phys. Rev. Appl. 9, 044036 (2018).
Poulin, D. et al. The Trotter step size required for accurate quantum simulation of quantum chemistry. Quantum Inf. Comput. 15, 361–384 (2015).
Peng, B. & Kowalski, K. Highly efficient and scalable compound decomposition of twoelectron integral tensor and its application in coupled cluster calculations. J. Chem. Theory Comput. 13, 4179 (2017).
Motta, M., Shee, J., Zhang, S. & Chan, G. K.L. Efficient ab initio auxiliaryfield quantum Monte Carlo calculations in gaussian bases via lowrank tensor decomposition. J. Chem. Theory Comput. 15, 3510–3521 (2019).
Reiher, M., Wiebe, N., Svore, K. M., Wecker, D. & Troyer, M. Elucidating reaction mechanisms on quantum computers. Proc. Natl Acad. Sci. 114, 7555 (2017).
Whitten, J. L. Coulombic potential energy integrals and approximations. J. Chem. Phys. 58, 4496 (1973).
Hohenstein, E. G. & Sherrill, C. D. Density fitting and Cholesky decomposition approximations in symmetryadapted perturbation theory: Implementation and application to probe the nature of ππ interactions in linear acenes. J. Chem. Phys. 132, 184111 (2010).
Beebe, N. H. & Linderberg, J. Simplifications in the generation and transformation of twoelectron integrals in molecular calculations. Int. J. Quantum Chem. 12, 683 (1977).
Koch, H., Sánchez de Merás, A. & Pedersen, T. B. Reduced scaling in electronic structure calculations using Cholesky decompositions. J. Chem. Phys. 118, 9481 (2003).
Aquilante, F. et al. MOLCAS 7: the next generation. J. Comput. Chem. 31, 224 (2010).
Purwanto, W., Krakauer, H., Virgus, Y. & Zhang, S. Assessing weak hydrogen binding on Ca^{+} centers: An accurate manybody study with large basis sets. J. Chem. Phys. 135, 164105 (2011).
Motta, M. & Zhang, S. Ab initio computations of molecular systems by the auxiliaryfield quantum Monte Carlo method. Wires Comput. Mol. Sci. 8, e1364 (2018).
Folkestad, S. D., Kjønstad, E. F. & Koch, H. An efficient algorithm for Cholesky decomposition of electron repulsion integrals. J. Chem. Phys. 150, 194112 (2019).
Petersson, G. A., Malick, D. K., Frisch, M. J. & Braunstein, M. The convergence of complete active space selfconsistentfield energies to the complete basis set limit. J. Chem. Phys. 123, 074111 (2005).
Wecker, D. et al. Solving strongly correlated electron models on a quantum computer. Phys. Rev. A 92, 062318 (2015b).
Niu, M. Y., Boixo, S., Smelyanskiy, V. N. & Neven, H. Universal quantum control through deep reinforcement learning. npj Quantum Inf. 5, 33 (2019).
Fowler, A. G., Mariantoni, M., Martinis, J. M. & Cleland, A. N. Surface codes: towards practical largescale quantum computation. Phys. Rev. A 86, 032324 (2012).
Bocharov, A., Roetteler, M. & Svore, K. M. Efficient synthesis of universal repeatuntilsuccess quantum circuits. Phys. Rev. Lett. 114, 080502 (2015).
Kivlichan, I. D. et al. Improved faulttolerant quantum simulation of condensedphase correlated electrons via Trotterization. Quantum 4, 296 (2020).
Arute, F. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505 (2019).
Childs, A. M., Maslov, D., Nam, Y., Ross, N. J. & Su, Y. Toward the first quantum simulation with quantum speedup. P. Natl Acad. Sci. 115, 9456 (2018).
Babbush, R. et al. Encoding electronic spectra in quantum circuits with linear T complexity. Phys. Rev. X 8, 041015 (2018b).
Shenvi, N. et al. The tensor hypercontracted parametric reduced density matrix algorithm: Coupledcluster accuracy with o(r^{4}) scaling. J. Chem. Phys. 139, 054110 (2013).
Matsuzawa, Y. & Kurashige, Y. Jastrowtype decomposition in quantum chemistry for lowdepth quantum circuits. J. Chem. Theory Comput. 16, 944 (2020).
Berry, D. W., Gidney, C., Motta, M., McClean, J. R. & Babbush, R. Qubitization of arbitrary basis quantum chemistry leveraging sparsity and low rank factorization. Quantum 3, 208 (2019).
von Burg, V. et al. Quantum computing enhanced computational catalysis. Preprint at: https://arxiv.org/pdf/2007.14460.pdf (2020).
Huggins, W. J. et al. Efficient and noise resilient measurements for quantum chemistry on nearterm quantum computers. npj Quantum Inf. 7, 23 (2021).
Yen, T.C. & Izmaylov, A. F. Cartan subalgebra approach to efficient measurements of quantum observables. Preprint at: https://arxiv.org/abs/2007.01234 (2020).
Sun, Q. et al. PySCF: the Pythonbased simulations of chemistry framework. Wires Comput. Mol. Sci. 8, e1340 (2018).
White, S. R. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 69, 2863 (1992).
Chan, G. K.L. & HeadGordon, M. Highly correlated calculations with a polynomial cost algorithm: A study of the density matrix renormalization group. J. Chem. Phys. 116, 4462 (2002).
Johnson, R. D. et al. NIST computational chemistry comparison and benchmark database. http://cccbdb.nist.gov/ (2018).
Acknowledgements
M.M. gratefully acknowledges Shiwei Zhang and James Shee for valuable interactions. M.M., Z.L., and G.K.C. (theoretical analysis, electronic structure calculations, drafting of the paper) were supported by NSF grant number 1839204. E.Y. (gate counts, electronic structure calculations) was supported by a Google graduate fellowship and a Google award to G.K.C. A.M. (drafting of the paper) was supported by NSF grant CBET CAREER number 1254213.
Author information
Authors and Affiliations
Contributions
R.B., G.K.C., and M.M. contributed to the conception of the project; M.M., J.R.M., Z.L., R.B., and G.K.C. contributed to the theoretical analysis; M.M., Z.L., E.Y., and G.K.C. contributed to the electronic structure calculations; M.M., J.R.M., E.Y., R.B., and G.K.C. contributed to the gate counts analysis. All authors contributed to the drafting of the paper.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Motta, M., Ye, E., McClean, J.R. et al. Low rank representations for quantum simulation of electronic structure. npj Quantum Inf 7, 83 (2021). https://doi.org/10.1038/s4153402100416z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s4153402100416z
This article is cited by

Efficient simulation of potential energy operators on quantum hardware: a study on sodium iodide (NaI)
Scientific Reports (2024)

NISQ computing: where are we and where do we go?
AAPPS Bulletin (2022)