PHYSICAL REVIEW RESEARCH 2, 023048 (2020) Efficient two-electron ansatz for benchmarking quantum chemistry on a quantum computer Scott E. Smart and David A. Mazziotti* Department of Chemistry and James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA (Received 18 December 2019; accepted 16 March 2020; published 17 April 2020) Quantum chemistry provides key applications for near-term quantum computing, but these are greatly complicated by the presence of noise. In this work we present an efficient ansatz for the computation of two- electron atoms and molecules within a hybrid quantum-classical algorithm. The ansatz exploits the fundamental structure of the two-electron system, treating the nonlocal and local degrees of freedom on the quantum and classical computers, respectively. Here the nonlocal degrees of freedom scale linearly with respect to basis-set size, giving a linear ansatz with only O(1) circuit preparations required for reduced state tomography. We implement this benchmark with error mitigation on two publicly available quantum computers, calculating + accurate dissociation curves for four- and six-qubit calculations of H2 and H3 . DOI: 10.1103/PhysRevResearch.2.023048 I. INTRODUCTION separation between the nonlocal and local fermionic degrees of freedom in the system [17], scaling linearly and polynomi- Quantum computers possess a natural affinity for quantum ally, respectively, and can be leveraged in a variational hybrid simulation and can transform exponentially scaling problems quantum-classical algorithm. The entangled nonlocal degrees into polynomial ones [1–3]. Quantum supremacy, the ability are treated on the quantum computer while the local degrees of a quantum computer to surpass its classical counterpart are treated on the classical computer, leading to an efficient on a designated task with lower asymptotic scaling, is poten- simulation of the system. tially realizable for the simulation of quantum many-electron For a quantum algorithm to exhibit quantum supremacy, systems [4,5]. Work over the previous decade has been to- obtaining the solution classically will be impractical except wards this goal with a focus on calculating the energy of for cases that are close to the classical limits of feasibil- small molecules and exploring strategies to leverage emerging ity [4,5]. For some problems such as prime factorization, the quantum technologies, especially those designed to correct solution can be quickly verified, but for many-body quantum or mitigate quantum errors [6–8]. In this paper we introduce systems this is not the case [18–22]. Possessing “easy” classi- an efficient ansatz for a two-electron quantum-mechanical cally solvable problems to implement and verify will be cru- system that can be employed as a benchmark for assessing cial to evaluate the performance of quantum devices and error the capabilities and accuracy of quantum computers. The twin mitigation schemes [23]. Our proposed quantum-classical hy- goals of the work are (i) to present a quantum-computing brid algorithm targets only the necessary entanglement needed benchmark based on the correlated but polynomial scaling on the quantum computer, scales linearly with respect to basis two-electron problem, solvable on classical computers, that size, and has O(1) circuit preparations, making it an ideal can be used to assess the accuracy of quantum computers benchmark for molecular simulation. We highlight this by and (ii) to develop an efficient ansatz for solving the two- + evaluating this ansatz through the computation of H and H electron problem on quantum computers, based on an effec- 2 3 on two generations of publicly available quantum devices. tive partitioning of the computational work between classical and quantum computers that is applicable to more general N-electron molecular systems. II. THEORY The two-electron density matrix (2DM) of any two- Because the representation of the 2DM in terms of electron system can be expressed as a functional of its one- the 1RDM and phase factors has been well studied else- electron reduced density matrix (1RDM) and a set of phase where [11,24–26], we present in Sec. II A only the aspects factors. This representation of the 2DM has important con- of the theory that are relevant to the quantum-classical hybrid nections to natural-orbital functional theories and geminal- algorithm in Sec. II B. We also discuss the preparation of the based theories in quantum chemistry [9–16]. It offers a natural linear-scaling ansatz in Sec. II C and practical error mitigation techniques in Sec. II D that are employed in the benchmarks in Sec. III. *Corresponding author: [email protected] A. Structure of the two-electron system Published by the American Physical Society under the terms of the For a two-electron system the energy is given as the trace Creative Commons Attribution 4.0 International license. Further of the Hamiltonian and the density matrix distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. E = Tr(HD), (1) 2643-1564/2020/2(2)/023048(8) 023048-1 Published by the American Physical Society SCOTT E. SMART AND DAVID A. MAZZIOTTI PHYSICAL REVIEW RESEARCH 2, 023048 (2020) where H and D are 2r × 2r, with 2r being the rank of the is simply the parity of the term that can easily be measured one-electron basis set and on a quantum computer. That is, we need the phase ξii where Dij = g g∗ (2) kl ij kl √ g˜ii = niξii . (8) in which the wave-function expansion coefficients gij are elements of the coefficient matrix G. From the antisymmetric nature of fermions, G must be a skew-symmetric matrix, In general, the phases can be measured through tomography ˜ and from a theorem by Zumino [26], G must have a block- on the quantum computer of certain terms of D requiring diagonal form G˜ with 2 × 2 matrices G : only O(1) additional circuit preparations. The details of the i specific ansatz for the tomography are discussed in the next G˜ = diag(G0, G1,...,Gr ), (3) section. After convergence criteria in the optimization of D˜ on the 0˜gii 0˜gii quantum computer are satisfied through gradient-free opti- Gi = = . (4) g˜ii 0 −g˜ii 0 mization (see Appendix A), we optimize the energy on the classical computer through one-body unitary transformations The block-diagonal form of G in Zumino’s theorem defines of the Hamiltonian. Specifically, we optimize the orbitals an orbital basis set with a natural pairing of the orbitals where through a series of Givens rotations. These complementary we denote the indices of an orbital and its pair by i and i , quantum and classical optimizations are sequentially repeated respectively. until the energy and 2DM converge. The 2DM in Zumino’s basis has only nonzero elements of the form ˜ ii = ∗ . C. Preparation of the efficient quantum ansatz Dkk g˜ii g˜kk (5) To create a state of the form in Eq. (3) on the quantum The 1RDM, containing the one-body information, can be computer, we need to implement double excitations from obtained from the 2DM by contraction, orbitals ii to kk. If we consider an initial wave function ˜ i ik ii ∗ G0 from a standard Hartree-Fock calculation, the ansatz to 1 ˜ = ˜ = ˜ = = = , Di Dik Dii g˜ii g˜ii ni ni (6) generate a generic G˜ is k 1 ˜ i = , r−1 D j 0 (7) ˜ = ¯i¯i ˜ , G exp tii G0 (9) where all 2DM elements in the contraction vanish except i=1 when k = i. Because the 1RDM is diagonal in Zumino’s basis set, we find that Zumino’s basis set is identically the ¯ = + ¯i¯i where i i 1 and tii is an anti-Hermitian, antisymmetric natural-orbital basis set and that the occupations ni are the two-body matrix with nonzero elements corresponding to an natural-orbital occupations. The paired orbitals i and i ,we excitation between i, i and (i + 1), (i + 1). The operators observe, have equal occupations ni and ni . For a system of acting on G˜ can be easily expressed in second quantiza- = 0 two electrons with Sz 0, these paired orbitals share the same tion as shown in the Appendixes. The ansatz is a subset of spatial component with different spin components, denoted by the unitary coupled cluster [32] or anti-Hermitian contracted α β convention by and . This decomposition can be viewed Schrödinger equation ansatz [33]. From there we perform a = as a particular case (N 2) of a more general result derived Jordan-Wigner transformation (though others may be utilized) by Schmidt [27] and later by Carlson and Keller [28]. The which yields an exponential of Pauli strings that are imple- importance of this decomposition as a quantum-computing mentable on a quantum device as strings of controlled-NOT ansatz for two electrons will be manifest below. (CNOT) gates [34]. Additionally, the implementation naturally requires only a nearest-neighbor connectivity among qubits. B. Variational hybrid algorithm Finally, the tomography of the state involves only the Hybrid quantum-classical algorithms with a variational measurement of the orbital occupations for a given qubit in the computational basis as well as the sign. Because there eigenvalue solver are among the most promising algorithms − for near-term applications [6,29–31]. For our approach we are only r 1 phase terms since the last phase is equivalent to a global phase, we only require the tomography of a utilize a variational eigensolver on the quantum device but jj linear number of sequential terms in D˜ of the form D˜ , apply it only to the optimization of the 2DM in the natural- ii = + ˜ jj ˜ ( j+2)(j+2) orbital basis set. Optimization of the natural orbitals by or- with j i 1.