# Enhancing the efficiency of quantum annealing via reinforcement: A path-integral Monte Carlo simulation of the quantum reinforcement algorithm

## TL;DR

This paper demonstrates through path-integral Monte Carlo simulations that quantum reinforcement can significantly improve the success probability of quantum annealing in solving hard combinatorial problems.

## Contribution

It introduces a local quantum reinforcement algorithm and shows its potential to enhance quantum annealing efficiency for constraint satisfaction problems.

## Key findings

- Quantum reinforcement increases success probability of quantum annealing.
- The local reinforcement algorithm performs well on XORSAT problems.
- Results suggest potential for larger problem sizes and classical optimization applications.

## Abstract

The standard quantum annealing algorithm tries to approach the ground state of a classical system by slowly decreasing the hopping rates of a quantum random walk in the configuration space of the problem, where the on-site energies are provided by the classical energy function. In a quantum reinforcement algorithm, the annealing works instead by increasing gradually the strength of the on-site energies according to the probability of finding the walker on each site of the configuration space. Here, by using the path-integral Monte Carlo simulations of the quantum algorithms, we show that annealing via reinforcement can significantly enhance the success probability of the quantum walker. More precisely, we implement a local version of the quantum reinforcement algorithm, where the system wave function is replaced by an approximate wave function using the local expectation values of the system. We use this algorithm to find solutions to a prototypical constraint satisfaction problem (XORSAT) close to the satisfiability to unsatisfiability phase transition. The study is limited to small problem sizes (a few hundreds of variables), nevertheless, the numerical results suggest that quantum reinforcement may provide a useful strategy to deal with other computationally hard problems and larger problem sizes even as a classical optimization algorithm.