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Resolution of the Relative Energies of the Benzyne Isomers on a Quantum Computer Using a Contracted Schr\"odinger Equation

Scott E. Smart, Jan-Niklas Boyn, David A. Mazziotti

Published Mar 11, 2021• 0 citations
OpenAlex

Abstract

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 ${\textrm C_6} {\textrm H_4}$ 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 $N$-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.

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quant-pharXiv:2604.02550•Apr 6, 2026
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.
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.
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.
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.
A recent advance in the theory of the contracted Schr\"odinger equation (CSE), in which only the anti-Hermitian part of the equation is solved, permits the direct determination of ground-state two-electron reduced density matrices (2-RDM's) that yield 95%--100% of the correlation energy of atoms and molecules [D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)]. Here we discuss in detail the anti-Hermitian contracted Schr\"odinger equation (ACSE) and its comparison to the CSE with regard to cumulant reconstruction of RDM's, the role of Nakatsuji's theorem, and the structure of the wave function. The ACSE is also formulated in the Heisenberg representation and related to canonical diagonalization. The solution of the ACSE is illustrated with a variety of molecules including H2O{\mathrm{H}}_{2}\mathrm{O}H2​O, CH2{\mathrm{CH}}_{2}CH2​, NH4+{{\mathrm{NH}}_{4}}^{+}NH4​+, HF\mathrm{HF}HF, and N2{\mathrm{N}}_{2}N2​, and potential energy and dipole-moment surfaces are computed for boron hydride in a polarized double-$\ensuremath{\zeta}$ basis set. The computed 2-RDM's very closely satisfy known NNN-representability conditions.
Two-electron reduced density matrices (2-RDMs) of many-electron molecules are directly determined without calculation of their wave functions by solving the anti-Hermitian contracted Schrödinger equation. Approximation of the 3-RDM in the anti-Hermitian contracted Schrödinger equation by a corrected cumulant expansion [Mazziotti, Phys. Rev. A 60, 3618 (1999)] permits the direct calculation of the energy and 2-RDM with many high-order correlation effects included. The method is illustrated for the molecules BeH2, H2O, NH3, CH4, and CO as well as the dissociation of BH. Correlation energies are obtained within 95%-100% of full-configuration interaction, and 2-RDMs very nearly satisfy known N-representability conditions.
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.