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arXiv:2403.00200 · Physical Review Research

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

Yoshitaka Inui, Edwin Ng, Y. Yamamoto

Published Nov 18, 2024• 4 citations
arXivHTMLPDF

Abstract

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

DOI
10.1103/physrevresearch.6.043151

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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
As a solver for non-deterministic polynomial time (NP)-hard combinatorial optimization problems, the coherent Ising machine (CIM) is in the early stages of research, and the potential of this innovative physical system will be developed. Here, we propose a speed-up coherent Ising machine with a squeezed feedback system, which we call S-CIM. We couple squeezed feedback pulses generated by the squeezed feedback system into the degenerate optical parametric oscillator (DOPO) network. Simulations indicate that quantum inseparability of the coupled DOPO network is further enhanced during the whole optimization process, and quantum fluctuations are significantly smaller around the oscillation threshold. Computation experiments are performed on MAX-CUT problems of order between 4 and 20000. Numerical results demonstrate that S-CIM increases the optimal normalized output by 2.27% and significantly reduces the optimal computation time by 75.12%.
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.
The Coherent Ising Machine (CIM) is a quantum network of optical parametric oscillators (OPOs) intended to find ground states of the Ising model. This is an NP-hard problem, related to several important minimization problems, including the max-cut graph problem. In order to enhance its potential performance, we analyze the coherent coupling strategy for the CIM in a highly quantum regime. To explore this limit, without assuming gaussianity, we employ accurate numerical simulations. Due to the inherent complexity of the system, the maximum network size is limited. While master equation methods can be used, their scalability diminishes rapidly for larger systems. Instead, we use Monte Carlo wave-function methods, which scale as the wave-function dimension, and use large numbers of samples. These simulations involve Hilbert spaces exceeding 10710^{7}107 dimensions. To evaluate success probabilities, we use quadrature probabilities. We demonstrate the potential for quantum computational advantage by reducing the time required to reach maximum success probability in a low-dissipation regime enabled by initial quantum superpositions and entanglement. Furthermore, we demonstrate that tailored time-dependent couplings can amplify these quantum effects. Comparisons with classical CIM models give evidence that quantum tunneling effects in this strong coupling limit can overcome trapping in false minima. This can greatly increase success rates, indicating a potential for quantum advantage. Finally, we perform a coherence analysis based on the state purity to examine the role of quantum coherence in CIM performance and to determine how state purity correlates with improved optimization outcomes.
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.
In this work, we study the detection of quantum phase transitions (QPTs) in coherent Ising machines (CIMs) through a spectral mapping between the one-dimensional XY spin model and a network of degenerate optical parametric oscillators (DOPOs). This exact correspondence reveals that the DOPO spectrum faithfully encodes the critical behavior of anisotropic and isotropic XY chains, while a strongly anisotropic regime provides a controlled near-transverse-field-Ising realization, with the energy gap closing precisely at the critical point. The ground-state energy density and its derivatives are analyzed, and the resulting singularities in magnetic susceptibility accurately pinpoint the quantum critical points of the corresponding spin model. These results show that CIMs not only serve as powerful platforms for solving combinatorial optimization problems but also provide a versatile tool for probing universal quantum critical phenomena, bridging quantum-spin models and photonic quantum systems.