# Message-passing algorithm of quantum annealing with nonstoquastic Hamiltonian

## TL;DR

This paper develops a message-passing algorithm to simulate quantum annealing with nonstoquastic Hamiltonians, enabling performance assessment without quantum Monte Carlo methods, and validates it against the replica method.

## Contribution

It introduces a novel message-passing algorithm for classical simulation of nonstoquastic Hamiltonians in quantum annealing, expanding the tools for analyzing complex quantum systems.

## Key findings

- The algorithm accurately assesses QA performance with nonstoquastic Hamiltonians.
- Classical simulation of certain nonstoquastic Hamiltonians is feasible at low temperatures.
- Validation shows consistency with the replica method results.

## Abstract

Quantum annealing (QA) is a generic method for solving optimization problems using fictitious quantum fluctuation. The current device performing QA involves controlling the transverse field; it is classically simulatable by using the standard technique for mapping the quantum spin systems to the classical ones. In this sense, the current system for QA is not powerful despite utilizing quantum fluctuation. Hence, we developed a system with a time-dependent Hamiltonian consisting of a combination of the formulated Ising model and the "driver" Hamiltonian with only quantum fluctuation. In the previous study, for a fully connected spin model, quantum fluctuation can be addressed in a relatively simple way. We proved that the fully connected antiferromagnetic interaction can be transformed into a fluctuating transverse field and is thus classically simulatable at sufficiently low temperatures. Using the fluctuating transverse field, we established several ways to simulate part of the nonstoquastic Hamiltonian on classical computers. We formulated a message-passing algorithm in the present study. This algorithm is capable of assessing the performance of QA with part of the nonstoquastic Hamiltonian having a large number of spins. In other words, we developed a different approach for simulating the nonstoquastic Hamiltonian without using the quantum Monte Carlo technique. Our results were validated by comparison to the results obtained by the replica method.