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A coherent Ising machine with quantum measurement and feedback control

Yoshitaka Haribara, Y. Yamamoto, Ken‐ichi Kawarabayashi, Shoko Utsunomiya

Published Jan 28, 2015• 5 citations

Abstract

Combinatorial optimization problems are computationally hard in general, but they are ubiquitous in our modern life. A coherent Ising machine (CIM) based on a multiple-pulse degenerate optical parametric oscillator (DOPO) is an alternative approach to solve these problems by a specialized physical computing system. To evaluate its potential performance, computational experiments are performed on maximum cut (MAX-CUT) problems against traditional algorithms such as semidefinite programming relaxation of Goemans-Williamson and simulated annealing by Kirkpatrick, et al. The numerical results empirically suggest that the almost constant computation time is required to obtain the reasonably accurate solutions of MAX-CUT problems on a CIM with the number of vertices up to $2 \times 10^4$ and the number of edges up to $10^8$.

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Combinatorial optimization problems are computationally hard in general, but they are ubiquitous in our modern life. A coherent Ising machine (CIM) based on a multiple-pulse degenerate optical parametric oscillator (DOPO) is an alternative approach to solve these problems by a specialized physical computing system. To evaluate its potential performance, computational experiments are performed on maximum cut (MAX-CUT) problems against traditional algorithms such as semidefinite programming relaxation of Goemans-Williamson and simulated annealing by Kirkpatrick, et al. The numerical results empirically suggest that the almost constant computation time is required to obtain the reasonably accurate solutions of MAX-CUT problems on a CIM with the number of vertices up to

2 \times 10^4

and the number of edges up to

10^8

.

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.

We report a coherent Ising machine (CIM), which is a network of degenerate optical parametric oscillators for simulating the Ising model. Using a long-distance fiber cavity that includes a telecom-band phase sensitive amplifier, we generated large number of degenerate optical parametric oscillator (DOPO) pulses multiplexed in time domain, which represent the Ising spins. By using a measurement and feedback scheme, we could implement all-to-all coupling among 2048 DOPOs. We experimentally confirmed that the CIM could find solutions to maximum cut problems for graphs with as many as 2000 nodes.

Coherent Ising machines (CIMs) are increasingly anticipated as a new computer that can solve combinatorial optimization problems. CIMs use degenerate optical parametric oscillator pulses as artificial spins. We constructed a 100,512-spin coherent Ising machine, the world's largest. Our CIM can find good solutions to maximum cut problems of 100,000-node graphs ~1000 times faster than simulated annealing. Furthermore, it provides both very good scores and a broad distribution when we operate it near the phase transition point.

We introduce the recent progress of coherent Ising machine (CIM), which finds solutions to complex combinatorial optimization problems using a network of degenerate optical parametric oscillators (DOPO). We describe our effort to generate time-multiplexed DOPOs using long-distance fiber cavities including telecom-band phase sensitive amplifiers [1-4]. We then present our experiments where we found solutions to complex combinatorial optimization problems using a CIM with measurement-feedback scheme [5]. Finally, a future perspective of the CIM project will briefly described.