# A Quantum Model for Coherent Ising Machine: Stochastic Differential Equations with Replicator Dynamics

## TL;DR

This paper develops a quantum theoretical framework for coherent Ising machines using stochastic differential equations and replicator dynamics, enabling simulation of quantum effects in Ising spin models.

## Contribution

It introduces a novel quantum model combining stochastic differential equations with replicator dynamics for coherent Ising machines.

## Key findings

- Simulated simple Ising spin models demonstrating quantum effects.
- Elucidated unique features of quantum coherent Ising machine.
- Provided a theoretical basis for future quantum optimization algorithms.

## Abstract

The quantum theory of coherent Ising machines, based on degenerate optical parametric oscillators and measurement-feedback circuits, is developed using the positive $P({\alpha},{\beta})$ representation of the density operator and the master equation. The theory is composed of the c-number stochastic differential equations for describing open dissipative quantum dynamics and the replicator dynamics equations for handling measurement-induced collapse of the density operator. We apply the present theory to simulate two simple Ising spin models and elucidate the unique features of this computing machine.