Verifiably exact solution of the electronic Schrödinger equation on quantum devices

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 Schrödinger equation. Rather than solve the Schrödinger equation directly, we solve its contraction over all electrons except two, known as the contracted Schrödinger 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 Schrödinger equation, the two-electron Ansatz provides a verifiably exact Ansatz for solving the many-electron Schrödinger equation. The exactness property contrasts with that of Ansätze 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 Schrödinger equation. We demonstrate the algorithm on both simulators and noisy quantum computers with ${\mathrm{H}}_2$ dissociation and the rectangle-to-square transition in ${\mathrm{H}}_4$.