Quantum Algorithm for a Convergent Series of Approximations towards the Exact Solution of the Lowest Eigenstates of a Hamiltonian

TL;DR

This paper introduces a quantum algorithm that converges towards the exact solution of the lowest eigenstates of a Hamiltonian, enabling efficient electronic structure calculations and potentially demonstrating quantum supremacy in chemistry.

Contribution

The paper presents a novel quantum algorithm for Hamiltonian eigenstate approximation using a convergent series, scalable to full configuration interaction calculations.

Findings

Scales as O(m^5) for molecular calculations, with full CI scaling as O(nm^5).

Enables implementation of various electronic structure approximations on quantum computers.

Potential to demonstrate quantum supremacy with low-order matrix-vector products.

Abstract

We present quantum algorithms, for Hamiltonians of linear combinations of local unitary operators, for Hamiltonian matrix-vector products and for preconditioning with the inverse of shifted reduced Hamiltonian operator that contributes to the diagonal matrix elements only. The algorithms implement a convergent series of approximations towards the exact solution of the full CI (configuration interaction) problem. The algorithm scales with O(m^5 ), with m the number of one-electron orbitals in the case of molecular electronic structure calculations. Full CI results can be obtained with a scaling of O(nm^5 ), with n the number of electrons and a prefactor on the order of 10 to 20. With low orders of Hamiltonian matrix-vector products, a whole repertoire of approximations widely used in modern electronic structure theory, including various orders of perturbation theory and/or truncated CI at different orders of excitations can be implemented for quantum computing for both routine and benchmark results at chemical accuracy. The lowest order matrix-vector product with preconditioning, basically the second-order perturbation theory, is expected to be a leading algorithm for demonstrating quantum supremacy for Ab Initio simulations, one of the most anticipated real world applications. The algorithm is also applicable for the hybrid variational quantum eigensolver.