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arXiv:2303.00758 · Physical Review A

Verifiably exact solution of the electronic Schrödinger equation on quantum devices

Scott E. Smart, David A. Mazziotti

Published Feb 5, 2024• 18 citations
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Abstract

Quantum computers have the potential for a significant speedup of molecular computations. However, existing algorithms have limitations; quantum phase estimation (QPE) is intractable on current hardware while variational quantum eigensolvers (VQE) are dependent upon approximate wave functions without guaranteed convergence. In this paper we present an algorithm that yields verifiably exact solutions of the many-electron Schr\"odinger equation. Rather than solve the Schr\"odinger equation directly, we solve its contraction over all electrons except two, known as the contracted Schr\"odinger equation (CSE). The CSE generates a wave-function Ansatz, constructed from an iterative product of nonunitary two-body transformations, whose energy gradient with respect to the two-body operator of the current iteration vanishes if and only if the CSE is satisfied. Because the CSE implies the Schr\"odinger equation, the two-electron Ansatz provides a verifiably exact Ansatz for solving the many-electron Schr\"odinger equation. The exactness property contrasts with that of Ans\"atze built from the product of unitary two-body transformations where the gradient---the residual of the anti-Hermitian part of the CSE (ACSE)---can vanish without implying a solution of the Schr\"odinger equation. We demonstrate the algorithm on both simulators and noisy quantum computers with ${\mathrm{H}}_{2}$ dissociation and the rectangle-to-square transition in ${\mathrm{H}}_{4}$.

DOI
10.1103/physreva.109.022802

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quant-pharXiv:2604.02550•Apr 6, 2026
Quantum computers have the potential for a significant speedup of molecular computations. However, existing algorithms have limitations; quantum phase estimation (QPE) is intractable on current hardware while variational quantum eigensolvers (VQE) are dependent upon approximate wave functions without guaranteed convergence. In this paper we present an algorithm that yields verifiably exact solutions of the many-electron Schr"odinger equation. Rather than solve the Schr"odinger equation directly, we solve its contraction over all electrons except two, known as the contracted Schr"odinger equation (CSE). The CSE generates a wave-function Ansatz, constructed from an iterative product of nonunitary two-body transformations, whose energy gradient with respect to the two-body operator of the current iteration vanishes if and only if the CSE is satisfied. Because the CSE implies the Schr"odinger equation, the two-electron Ansatz provides a verifiably exact Ansatz for solving the many-electron Schr"odinger equation. The exactness property contrasts with that of Ans"atze built from the product of unitary two-body transformations where the gradient---the residual of the anti-Hermitian part of the CSE (ACSE)---can vanish without implying a solution of the Schr"odinger equation. We demonstrate the algorithm on both simulators and noisy quantum computers with H2{\mathrm{H}}_{2}H2​ dissociation and the rectangle-to-square transition in H4{\mathrm{H}}_{4}H4​.
A contracted quantum eigensolver (CQE) finds a solution to the many-electron Schrödinger equation by solving its integration (or contraction) to the two-electron space─a contracted Schrödinger equation (CSE)─on a quantum computer. When applied to the anti-Hermitian part of the CSE (ACSE), the CQE iterations optimize the wave function, with respect to a general product ansatz of two-body exponential unitary transformations that can exactly solve the Schrödinger equation. In this work, we accelerate the convergence of the CQE and its wave function ansatz via tools from classical optimization theory. By treating the CQE algorithm as an optimization in a local parameter space, we can apply quasi-second-order optimization techniques, such as quasi-Newton approaches or nonlinear conjugate gradient approaches. Practically, these algorithms result in superlinear convergence of the wave function to a solution of the ACSE. Convergence acceleration is important because it can both minimize the accumulation of noise on near-term intermediate-scale quantum (NISQ) computers and achieve highly accurate solutions on future fault-tolerant quantum devices. We demonstrate the algorithm, as well as some heuristic implementations relevant for cost-reduction considerations, comparisons with other common methods such as variational quantum eigensolvers, and a Fermionic-encoding-free form of the CQE.
Electronic excited states of molecules are central to many physical and chemical processes, and yet they are typically more difficult to compute than ground states. In this paper we leverage the advantages of quantum computers to develop an algorithm for the highly accurate calculation of excited states. We solve a contracted Schr\"odinger equation (CSE)---a contraction (projection) of the Schr\"odinger equation onto the space of two electrons---whose solutions correspond identically to the ground and excited states of the Schr\"odinger equation. While recent quantum algorithms for solving the CSE, known as contracted quantum eigensolvers (CQEs), have focused on ground states, we develop a CQE based on the variance that is designed to optimize rapidly to a ground or excited state. We apply the algorithm to compute the ground and excited states of H2,H4{\mathrm{H}}_{2}, {\mathrm{H}}_{4}H2​,H4​, and BH.
A contracted quantum eigensolver (CQE) finds a solution to the many-electron Schrödinger equation by solving its integration (or contraction) to the two-electron space─a contracted Schrödinger equation (CSE)─on a quantum computer. When applied to the anti-Hermitian part of the CSE (ACSE), the CQE iterations optimize the wave function, with respect to a general product ansatz of two-body exponential unitary transformations that can exactly solve the Schrödinger equation. In this work, we accelerate the convergence of the CQE and its wave function ansatz via tools from classical optimization theory. By treating the CQE algorithm as an optimization in a local parameter space, we can apply quasi-second-order optimization techniques, such as quasi-Newton approaches or nonlinear conjugate gradient approaches. Practically, these algorithms result in superlinear convergence of the wave function to a solution of the ACSE. Convergence acceleration is important because it can both minimize the accumulation of noise on near-term intermediate-scale quantum (NISQ) computers and achieve highly accurate solutions on future fault-tolerant quantum devices. We demonstrate the algorithm, as well as some heuristic implementations relevant for cost-reduction considerations, comparisons with other common methods such as variational quantum eigensolvers, and a Fermionic-encoding-free form of the CQE.
We present an exact Ansatz for the eigenstate problem of mixed fermion-boson systems that can be implemented on quantum devices. Based on a generalization of the electronic contracted Schrödinger equation (CSE), our approach guides a trial wave function to the ground state of any arbitrary mixed Hamiltonian by directly measuring residuals of the mixed CSE on a quantum device. Unlike density functional and coupled cluster theories applied to electron-phonon or electron-photon systems, the accuracy of our approach is not limited by the unknown exchange-correlation functional or the uncontrolled form of the exponential Ansatz. To test the performance of the method, we study the Tavis-Cummings model, commonly used in polaritonic quantum chemistry. Our results demonstrate that the CSE is a powerful tool in the development of quantum algorithms for solving general fermion-boson many-body problems.
Quantum computers potentially have an exponential advantage over classical computers for the quantum simulation of many-fermion quantum systems. Nonetheless, fermions are more expensive to simulate than bosons due to the fermionic encoding---a mapping by which the qubits are encoded with fermion statistics. Here we generalize the contracted quantum eigensolver (CQE) to avoid fermionic encoding of the wave function. In contrast to the variational quantum eigensolver, the CQE solves for a many-fermion stationary state by minimizing the contraction (projection) of the Schr\"odinger equation onto two fermions. We avoid fermionic encoding of the wave function by contracting the Schr\"odinger equation onto an unencoded pair of particles. Solution of the resulting contracted equation by a series of unencoded two-body exponential transformations generates an unencoded wave function from which the energy and two-fermion reduced density matrix (2-RDM) can be computed. We apply the unencoded and the encoded CQE algorithms to the hydrogen fluoride molecule, the dissociation of oxygen O2{\mathrm{O}}_{2}O2​, and a series of hydrogen chains. Both algorithms show comparable convergence towards the exact ground-state energies and 2-RDMs, but the unencoded algorithm has computational advantages in terms of state preparation and tomography.
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.