Diffusion Monte Carlo approach versus adiabatic computation for local Hamiltonians

TL;DR

This paper demonstrates that diffusion Monte Carlo can fail to efficiently simulate quantum adiabatic optimization even for local Hamiltonians, supported by a new counterexample and empirical results showing quantum advantage.

Contribution

It introduces a new 6-local counterexample and empirically compares diffusion Monte Carlo with quantum optimization on permutation-symmetric problems.

Findings

Diffusion Monte Carlo fails for certain 6-local Hamiltonians.

Quantum optimization outperforms diffusion Monte Carlo on tested problems.

Abstract

Most research regarding quantum adiabatic optimization has focused on stoquastic Hamiltonians, whose ground states can be expressed with only real, nonnegative amplitudes. This raises the question of whether classical Monte Carlo algorithms can efficiently simulate quantum adiabatic optimization with stoquastic Hamiltonians. Recent results have given counterexamples in which path integral and diffusion Monte Carlo fail to do so. However, most adiabatic optimization algorithms, such as for solving MAX-k-SAT problems, use k-local Hamiltonians, whereas our previous counterexample for diffusion Monte Carlo involved n-body interactions. Here we present a new 6-local counterexample which demonstrates that even for these local Hamiltonians there are cases where diffusion Monte Carlo cannot efficiently simulate quantum adiabatic optimization. Furthermore, we perform empirical testing of diffusion Monte Carlo on a standard well-studied class of permutation-symmetric tunneling problems and similarly find large advantages for quantum optimization over diffusion Monte Carlo.