Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1706.02034 · Physical Review A

Quantum model for coherent Ising machines: Stochastic differential equations with replicator dynamics

Taime Shoji, Kazuyuki Aihara, Y. Yamamoto

Published Nov 14, 2017• 30 citations
arXivHTMLPDF

Abstract

The quantum theory of coherent Ising machines, based on degenerate optical parametric oscillators and measurement-feedback circuits, is developed using the positive $P(\ensuremath{\alpha},\ensuremath{\beta})$ 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.

DOI
10.1103/physreva.96.053833

Related papers

Computational Principle and Performance Evaluation of Coherent Ising Machine Based on Degenerate Optical Parametric Oscillator Network

Yoshitaka Haribara, Shoko Utsunomiya, Y. Yamamoto

We present the operational principle of a coherent Ising machine (CIM) based on a degenerate optical parametric oscillator (DOPO) network. A quantum theory of CIM is formulated, and the computational ability of CIM is evaluated by numerical simulation based on c-number stochastic differential equations. We also discuss the advanced CIM with quantum measurement-feedback control and various problems which can be solved by CIM.

Entropy•Apr 19, 2016•70 citations

Quantum model for coherent Ising machines: Discrete-time measurement feedback formulation

Atsushi Yamamura, Kazuyuki Aihara, Y. Yamamoto

Recently, the coherent Ising machine (CIM) as a degenerate optical parametric oscillator (DOPO) network has been researched to solve Ising combinatorial optimization problems. We formulate a theoretical model for the CIM with discrete-time measurement feedback processes, and perform numerical simulations for the simplest network, composed of two degenerate optical parametric oscillator pulses with the antiferromagnetic mutual coupling. We evaluate the extent to which quantum coherence exists during the optimization process.

arXiv:1706.02454•Physical Review A•Nov 14, 2017•47 citations

Truncated Wigner theory of coherent Ising machines based on degenerate optical parametric oscillator network

Daiki Maruo, Shoko Utsunomiya, Y. Yamamoto

Abstract We present the quantum theory of coherent Ising machines based on networks of degenerate optical parametric oscillators (DOPOs). In a simple model consisting of two coupled DOPOs, both positive- P representation and truncated Wigner representation predict quantum correlation and inseparability between the two DOPOs in spite of the open-dissipative nature of the system. Here, we apply the truncated Wigner representation method to coherent Ising machines with thermal, vacuum, and squeezed reservoir fields. We find that the probability of finding the ground state of a one-dimensional Ising model increases substantially as a result of reducing excess thermal noise and squeezing the incident vacuum fluctuation on the out-coupling port.

arXiv:1605.08655•Physica Scripta•Jul 26, 2016•46 citations

Coherent Ising machine with quantum feedback: The total and conditional master equation methods

Simon Kiesewetter, P. D. Drummond

We give a detailed theoretical derivation of the master equation for the coherent Ising machine. This is a quantum computational network with feedback, that approximately solves NP-hard combinatoric problems, including the traveling salesman problem and various extensions and analogs. There are two possible types of master equation, either conditional on the feedback current or unconditional. We show that both types can be accurately simulated in a scalable way using stochastic equations in the positive-P phase-space representation. This depends on the nonlinearity present, and we use parameter values that are typical of current experiments. While the two approaches are in excellent agreement, they are not equivalent with regard to efficiency. We find that unconditional simulation has much greater efficiency, and is more scalable to large sizes. This is a case where too much knowledge is a dangerous thing. Conditioning the simulations on the feedback current is not essential to determining the success probability, but it greatly increases the computational complexity. To illustrate the speed improvements obtained with the unconditional approach, we carry out full quantum simulations of the master equation with up to 1000 nodes.

arXiv:2203.02447•Physical Review A•Aug 12, 2022•14 citations

Skewed-Gaussian model of small-photon-number coherent Ising machines

Yoshitaka Inui, Edwin Ng, Y. Yamamoto

A Gaussian quantum theory of bosonic modes has been widely used to describe quantum optical systems, including coherent Ising machines (CIMs) that consist of χ ( 2 ) degenerate optical parametric oscillators (DOPOs) as nonlinear elements. However, Gaussian models have been thought to be invalid in the extremely strong-gain-saturation limit. Here, we develop an extended Gaussian model including two third-order fluctuation products, 〈 δ X ̂ 3 〉 and 〈 δ X ̂ δ P ̂ 2 〉 , which we call self-skewness and cross-skewness, respectively. This new model which we call the skew-Gaussian model more precisely replicates the success probability predicted by the quantum master equation (QME), relative to Gaussian models. We also discuss the impact of skew variables on the performance of CIMs. Published by the American Physical Society 2024

arXiv:2403.00200•Physical Review Research•Nov 18, 2024•4 citations

Coherent Ising Machine - Optical Neural Network operating at the Quantum Limit

Y. Yamamoto

We will present the basic concept, operational principle and experimental performance of a novel computing machine with the network of degenerate optical parametric oscillators. The developed machine has 2048 qubits with all-to-all connections.

•Jan 1, 2018•2 citations
The quantum theory of coherent Ising machines, based on degenerate optical parametric oscillators and measurement-feedback circuits, is developed using the positive P(\ensuremathα,\ensuremathβ)P(\ensuremath{\alpha},\ensuremath{\beta})P(\ensuremathα,\ensuremathβ) 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.
Recently, the coherent Ising machine (CIM) as a degenerate optical parametric oscillator (DOPO) network has been researched to solve Ising combinatorial optimization problems. We formulate a theoretical model for the CIM with discrete-time measurement feedback processes, and perform numerical simulations for the simplest network, composed of two degenerate optical parametric oscillator pulses with the antiferromagnetic mutual coupling. We evaluate the extent to which quantum coherence exists during the optimization process.
Abstract We present the quantum theory of coherent Ising machines based on networks of degenerate optical parametric oscillators (DOPOs). In a simple model consisting of two coupled DOPOs, both positive- P representation and truncated Wigner representation predict quantum correlation and inseparability between the two DOPOs in spite of the open-dissipative nature of the system. Here, we apply the truncated Wigner representation method to coherent Ising machines with thermal, vacuum, and squeezed reservoir fields. We find that the probability of finding the ground state of a one-dimensional Ising model increases substantially as a result of reducing excess thermal noise and squeezing the incident vacuum fluctuation on the out-coupling port.
We give a detailed theoretical derivation of the master equation for the coherent Ising machine. This is a quantum computational network with feedback, that approximately solves NP-hard combinatoric problems, including the traveling salesman problem and various extensions and analogs. There are two possible types of master equation, either conditional on the feedback current or unconditional. We show that both types can be accurately simulated in a scalable way using stochastic equations in the positive-P phase-space representation. This depends on the nonlinearity present, and we use parameter values that are typical of current experiments. While the two approaches are in excellent agreement, they are not equivalent with regard to efficiency. We find that unconditional simulation has much greater efficiency, and is more scalable to large sizes. This is a case where too much knowledge is a dangerous thing. Conditioning the simulations on the feedback current is not essential to determining the success probability, but it greatly increases the computational complexity. To illustrate the speed improvements obtained with the unconditional approach, we carry out full quantum simulations of the master equation with up to 1000 nodes.
A Gaussian quantum theory of bosonic modes has been widely used to describe quantum optical systems, including coherent Ising machines (CIMs) that consist of χ ( 2 ) degenerate optical parametric oscillators (DOPOs) as nonlinear elements. However, Gaussian models have been thought to be invalid in the extremely strong-gain-saturation limit. Here, we develop an extended Gaussian model including two third-order fluctuation products, 〈 δ X ̂ 3 〉 and 〈 δ X ̂ δ P ̂ 2 〉 , which we call self-skewness and cross-skewness, respectively. This new model which we call the skew-Gaussian model more precisely replicates the success probability predicted by the quantum master equation (QME), relative to Gaussian models. We also discuss the impact of skew variables on the performance of CIMs. Published by the American Physical Society 2024