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arXiv:2106.11972 · The Journal of Chemical Physics

Quantum-classical hybrid algorithm for the simulation of all-electron correlation

Jan-Niklas Boyn, Лыхин Александр Олегович, Scott E. Smart, Laura Gagliardi, David A. Mazziotti

Published Dec 28, 2021• 32 citations
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Abstract

While chemical systems containing hundreds to thousands of electrons remain beyond the reach of quantum devices, hybrid quantum-classical algorithms present a promising pathway toward a quantum advantage. Hybrid algorithms treat the exponentially scaling part of the calculation-the static correlation-on the quantum computer and the non-exponentially scaling part-the dynamic correlation-on the classical computer. While a variety of algorithms have been proposed, the dependence of many methods on the total wave function limits the development of easy-to-use classical post-processing implementations. Here, we present a novel combination of quantum and classical algorithms, which computes the all-electron energy of a strongly correlated molecular system on the classical computer from the 2-electron reduced density matrix (2-RDM) evaluated on the quantum device. Significantly, we circumvent the wave function in the all-electron calculations by using density matrix methods that only require input of the statically correlated 2-RDM. Although the algorithm is completely general, we test it with two classical density matrix methods, the anti-Hermitian contracted Schrödinger equation (ACSE) and multiconfiguration pair-density functional theories, using the recently developed quantum ACSE method for simulating the statically correlated 2-RDM. We obtain experimental accuracy for the relative energies of all three benzyne isomers and thereby demonstrate the ability of the developed algorithm to achieve chemically relevant and accurate results on noisy intermediate-scale quantum devices.

DOI
10.1063/5.0074842

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While chemical systems containing hundreds to thousands of electrons remain beyond the reach of quantum devices, hybrid quantum-classical algorithms present a promising pathway toward a quantum advantage. Hybrid algorithms treat the exponentially scaling part of the calculation-the static correlation-on the quantum computer and the non-exponentially scaling part-the dynamic correlation-on the classical computer. While a variety of algorithms have been proposed, the dependence of many methods on the total wave function limits the development of easy-to-use classical post-processing implementations. Here, we present a novel combination of quantum and classical algorithms, which computes the all-electron energy of a strongly correlated molecular system on the classical computer from the 2-electron reduced density matrix (2-RDM) evaluated on the quantum device. Significantly, we circumvent the wave function in the all-electron calculations by using density matrix methods that only require input of the statically correlated 2-RDM. Although the algorithm is completely general, we test it with two classical density matrix methods, the anti-Hermitian contracted Schrödinger equation (ACSE) and multiconfiguration pair-density functional theories, using the recently developed quantum ACSE method for simulating the statically correlated 2-RDM. We obtain experimental accuracy for the relative energies of all three benzyne isomers and thereby demonstrate the ability of the developed algorithm to achieve chemically relevant and accurate results on noisy intermediate-scale quantum devices.
The simulation of strongly correlated many-electron systems is one of the most promising applications for near-term quantum devices. Here we develop and apply a family of quantum simulation algorithms for electronic structure, based on the integration (or contraction) of the Schr\odinger equation onto the two-electron space. We showcase the family of algorithms by resolving the energy splittings of the ortho-, meta-, and para-isomers of benzyne C6H4{\textrm C_6} {\textrm H_4}C6​H4​ including the strongly correlated biradical para-benzyne. The quantum solution of the anti-Hermitian part of the contracted Schr\odinger equation (qACSE) provides a scalable approach with low circuit depth and few variational parameters that has its foundations in two-electron reduced density matrix (2-RDM) theory. Error in the 2-RDM is mitigated through projection of the 2-RDM measured by tomography onto the approximate set of physically realistic 2-RDMs defined by the 2-positive NNN-representability conditions. The relative energies exhibit single-digit millihartree errors, and the computed natural-orbital occupations capture significant differences in the electron correlation of the isomers including the biradical nature of the {\em para}-isomer.
Quantum algorithms for molecular electronic structure have been developed with lower computational scaling than their classical counterparts, but emerging quantum hardware is far from being capable of the coherence, connectivity, and gate errors required for their experimental realization. Here we propose a class of quantum-classical hybrid algorithms that computes the energy from a two-electron reduced density matrix (2-RDM). The 2-RDM is constrained by NNN-representability conditions, constraints for representing an NNN-electron wave function, which mitigate noise from the quantum circuit. We compute the strongly correlated dissociation of doublet H3{\mathrm{H}}_{3}H3​ into three hydrogen atoms. The hybrid quantum-classical computer matches the energies from full configuration interaction to 0.1 kcal/mol, one-tenth of ``chemical accuracy,'' even in the strongly correlated limit of dissociation. Furthermore, the spatial locality of the computed one-electron RDM reveals that the quantum computer accurately predicts the Mott metal-insulator transition.
The simulation of strongly correlated many-electron systems is one of the most promising applications for near-term quantum devices. Here we use a class of eigenvalue solvers [presented in Smart and Mazziotti, Phys. Rev. Lett. 126, 070504 (2021)] in which a contraction of the Schr\"odinger equation is solved for the two-electron reduced density matrix (2-RDM) to resolve the energy splittings of the ortho-, meta-, and para-isomers of benzyne C6H4{\text{C}}_{6}{\text{H}}_{4}C6​H4​. In contrast to the traditional variational quantum eigensolver, the contracted quantum eigensolver can solve an integration (or contraction) of the many-electron Schr\"odinger equation onto the two-electron space. The quantum solution of the anti-Hermitian part of the contracted Schr\"odinger equation provides a scalable approach with few variational parameters that has its foundations in 2-RDM theory. Experimentally, a variety of error-mitigation strategies enable the calculation, including a linear shift in the 2-RDM targeting the iterative nature of the algorithm as well as a projection of the 2-RDM onto the convex set of approximately NNN-representable 2-RDMs defined by the 2-positive NNN-representability conditions. The relative energies exhibit single-digit millihartree errors, capturing a large part of the electron correlation energy, and the computed natural orbital occupations reflect the significant differences in the electron correlation of the isomers.
Accurately describing strong electron correlation in complex systems remains a prominent challenge in computational chemistry as near-term quantum algorithms treating total correlation often require prohibitively deep circuits. Here we present a hybrid strategy combining the Variational Quantum Eigensolver with Multiconfiguration Pair-Density Functional Theory to efficiently decouple correlation effects. This approach confines static correlation to a compact multireference quantum state while recovering dynamic correlation through a classical on-top density functional using reduced-density information. By enabling self-consistent orbital optimization, the method significantly reduces quantum resource overheads without sacrificing physical rigor. We demonstrate chemical accuracy on standard benchmarks by reproducing C2_22​ equilibrium bond lengths and benzene excitation energies with mean absolute errors of 0.006 Å and 0.048 eV respectively. Most notably, for the strongly correlated Cr2_22​ dimer requiring a large complete active space (48e, 42o), the framework yields a bound potential-energy curve and recovers qualitative dissociation behavior despite realistic hardware noise. These results establish that separating correlation types provides a practical route to reliable predictions on near-term quantum hardware.
Efficient simulation of strongly correlated electrons has become a routine tool in molecular electronic structure theory due to recent advances in approximate configuration interaction (CI) techniques. Nonetheless, the quantitative and predictive description of molecular electronic states remains a significant challenge due to the difficulty of computing all-electron correlation beyond CI. Here, we describe a new open-source implementation of the anti-Hermitian contracted Schrödinger equation (ACSE) for use in accurate simulation of all-electron correlation in molecules. In contrast to standard approaches via multireference perturbation theory, the scaling of the ACSE does not depend on the complexity of the strongly correlated reference wavefunction. Furthermore, the ACSE employs the exact electronic Hamiltonian, rather than an approximate perturbative Hamiltonian. Our benchmark results demonstrate good accuracy for main group and transition metal systems, in weakly and strongly correlated regimes, with various basis sets, and for ground and excited states. The results suggest that the ACSE has potential as a scalable and robust technique for simulating all-electron correlation in molecular ground and excited states.
Two-electron reduced density matrices (2-RDMs) have recently been directly calculated by solving the anti-Hermitian contracted Schr\"odinger equation (ACSE) to obtain 959595--100100\phantom{\rule{0.3em}{0ex}}%100 of the ground-state correlation energy of atoms and molecules with the accuracy increasing with the size of the one-electron basis set [Mazziotti, Phys. Rev. Lett. 97, 143002 (2006).] In this paper, the ACSE method is extended to treat strong multireference correlation effects that are often important at nonequilibrium molecular geometries. While previous ACSE calculations have employed an initial 2-RDM from the Hartree-Fock method, we initialize the solution of the ACSE with a 2-RDM guess from a multiconfiguration self-consistent field calculation. Applications are made to multireference correlation in the potential energy surfaces of the molecules HF\mathrm{HF}HF, H2O{\mathrm{H}}_{2}\mathrm{O}H2​O, and C2{\mathrm{C}}_{2}C2​ in polarized valence double-zeta basis sets while N2{\mathrm{N}}_{2}N2​ is treated in polarized valence double- and triple-zeta basis sets. Accurate ground-state energies and 1-RDM occupation numbers are obtained at both equilibrium and nonequilibrium geometries where the energies are within a few millihartrees of those from full configuration interaction.