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arXiv:1605.08655 · Physica Scripta

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

Daiki Maruo, Shoko Utsunomiya, Y. Yamamoto

Published Jul 26, 2016• 46 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/0031-8949/91/8/083010

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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.
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We present analytical and numerical simulation results for squeezing, entanglement, and quantum discord in a dissipatively coupled coherent Ising machine (CIM). Both analytical solutions and numerical simulation results, which are obtained with positive-PPP, truncated-Wigner, and truncated-Husimi representations for the density operator, predict the presence of entanglement and quantum discord, below and above the threshold of CIM. The entanglement criteria and quantum discord are evaluated as a function of the dissipative coupling strength relative to the background loss. For coupled two DOPOs, while entanglement disappears as the background loss exceeds the Ising coupling strength, the quantum discord remains finite even with a large linear loss. For one-dimensional lattice of DOPOs, while entanglement disappears for DOPO pair with large distance, quantum discord remains finite.
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%.
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.
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