Classical Algorithms for Constant Approximation of the Ground State Energy of Local Hamiltonians

TL;DR

This paper presents classical algorithms capable of approximating the ground state energy of local Hamiltonians efficiently, matching quantum algorithm complexities when a guiding state is available and providing new bounds without such guidance.

Contribution

It introduces classical algorithms for constant approximation of ground state energy that match quantum complexities with a guiding state and achieve new bounds without guidance.

Findings

Classical algorithms match quantum complexities with guiding states.

New classical bounds for standard local Hamiltonian problem.

Efficient algorithms for higher-precision approximation.

Abstract

We construct classical algorithms computing an approximation of the ground state energy of an arbitrary $k$-local Hamiltonian acting on $n$ qubits. We first consider the setting where a good ``guiding state'' is available, which is the main setting where quantum algorithms are expected to achieve an exponential speedup over classical methods. We show that a constant approximation (i.e., an approximation with constant relative accuracy) of the ground state energy can be computed classically in $\mathrm{poly}\left(1/\chi,n\right)$ time and $\mathrm{poly}(n)$ space, where $\chi$ denotes the overlap between the guiding state and the ground state (as in prior works in dequantization, we assume sample-and-query access to the guiding state). This gives a significant improvement over the recent classical algorithm by Gharibian and Le Gall (SICOMP 2023), and matches (up a to polynomial overhead) both the time and space complexities of quantum algorithms for constant approximation of the ground state energy. We also obtain classical algorithms for higher-precision approximation. For the setting where no guided state is given (i.e., the standard version of the local Hamiltonian problem), we obtain a classical algorithm computing a constant approximation of the ground state energy in $2^{O(n)}$ time and $\mathrm{poly}(n)$ space. To our knowledge, before this work it was unknown how to classically achieve these bounds simultaneously, even for constant approximation. We also discuss complexity-theoretic aspects of our results.