# Fair sampling of ground-state configurations of binary optimization problems

## TL;DR

This paper introduces a novel sampling method combining parallel tempering Monte Carlo and isoenergetic cluster moves to fairly sample ground states in binary optimization problems, addressing biases in existing heuristics.

## Contribution

The paper presents a new approach for unbiased sampling of degenerate ground states in binary optimization, applicable to quantum annealing and classical heuristics.

## Key findings

- Method effectively samples degenerate states with nearly equal probability.
- Applicable to Ising spin glasses and quantum annealers like D-Wave.
- Includes a heuristic for estimating the number of solutions.

## Abstract

Although many efficient heuristics have been developed to solve binary optimization problems, these typically produce correlated solutions for degenerate problems. Most notably, transverse-field quantum annealing - the heuristic employed in current commercially-available quantum annealing machines - has been shown to often be exponentially biased when sampling the solution space. Here we present an approach to sample ground-state (or low-energy) configurations for binary optimization problems. The method samples degenerate states with almost equal probability and is based on a combination of parallel tempering Monte Carlo with isoenergetic cluster moves. We illustrate the approach using two-dimensional Ising spin glasses, as well as spin glasses on the D-Wave Systems Inc. quantum annealer chimera topology. In addition, a simple heuristic to approximate the number of solutions of a degenerate problem is introduced.