Adiabatic optimization versus diffusion Monte Carlo

TL;DR

This paper investigates the limitations of diffusion Monte Carlo algorithms in simulating stoquastic adiabatic optimization and introduces a new method, Substochastic Monte Carlo, which performs well on MAX-k-SAT problems.

Contribution

The paper identifies fundamental obstructions in diffusion Monte Carlo algorithms and proposes Substochastic Monte Carlo as an effective alternative for classical optimization.

Findings

Diffusion Monte Carlo algorithms face obstructions in simulating stoquastic adiabatic evolution.

Substochastic Monte Carlo performs well on MAX-k-SAT problems, competitive with existing heuristics.

The new method offers a practical classical approach for certain optimization problems.

Abstract

Most experimental and theoretical studies of adiabatic optimization use stoquastic Hamiltonians, whose ground states are expressible using only real nonnegative amplitudes. This raises a question as to whether classical Monte Carlo methods can simulate stoquastic adiabatic algorithms with polynomial overhead. Here, we analyze diffusion Monte Carlo algorithms. We argue that, based on differences between L1 and L2 normalized states, these algorithms suffer from certain obstructions preventing them from efficiently simulating stoquastic adiabatic evolution in generality. In practice however, we obtain good performance by introducing a method that we call Substochastic Monte Carlo. In fact, our simulations are good classical optimization algorithms in their own right, competitive with the best previously known heuristic solvers for MAX-k-SAT at k=2,3,4.