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 ${\mathrm{H}}_{4}$ .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 ${\mathrm{H}}_{4}$.

arXiv:2303.00758Physical Review AFeb 5, 202418 citations

Numerical solutions of the Schrödinger equation

Peter Senn

American Journal of PhysicsSep 1, 19926 citations

Approximate and numerical solutions of the Schrödinger equation

Eric A. B. Cole

Springer eBooksJan 1, 20090 citations

An iterative solution of Schrodinger's equation

M. Cohen

The energy function E( lambda ) of a quantum system with Hamiltonian H( lambda )=H 0 + lambda H 1 is determined by an analytical iterative procedure which yields a rich variety of functional forms. The wavefunction iterates include the solutions of Rayleigh-Schrodinger perturbation theory as well as some additional functions, all of which may be obtained sequentially by standard analytical techniques. At each step, an arbitrary ( lambda -dependent) constant may be chosen freely so as to improve the results of low-order iterates, thereby enhancing the convergence of the procedure.The energy function E( lambda ) of a quantum system with Hamiltonian H( lambda )=H 0 + lambda H 1 is determined by an analytical iterative procedure which yields a rich variety of functional forms. The wavefunction iterates include the solutions of Rayleigh-Schrodinger perturbation theory as well as some additional functions, all of which may be obtained sequentially by standard analytical techniques. At each step, an arbitrary ( lambda -dependent) constant may be chosen freely so as to improve the results of low-order iterates, thereby enhancing the convergence of the procedure.

Journal of Physics A Mathematical and GeneralJun 1, 19842 citations

Solutions to the Schrödinger Equation

WORLD SCIENTIFIC eBooksSep 23, 20140 citations

{\mathrm{H}}_{2}

dissociation and the rectangle-to-square transition in

{\mathrm{H}}_{4}