Observing a Phase Transition on a Coherent Ising Machine

Abstract

A coherent Ising machine (CIM) is known to deliver the low-energy states of the Ising model. Here, we investigate how well the CIM simulates the thermodynamic properties of a two-dimensional square-lattice Ising model. Assuming that the spin sets sampled by the CIM can be regarded as a canonical ensemble, we estimate the effective temperature of the spins represented by degenerate optical parametric oscillator pulses by using maximum likelihood estimation. With the obtained temperature, we confirmed that the thermodynamic quantities obtained with the CIM exhibited a phase-transition-like behavior that better matches the analytical and numerical results than the mean field one.

Thermodynamic quantities of two-dimensional Ising models obtained by noisy mean field annealing and coherent Ising machine

“…The deviation between these two definitions of specific heat at low temperature indicates that the spins delivered by the CIM are not perfectly sampled from an ideal canonical ensemble. At high temperature, the canonical sampling assumption is satisfied, which is nontrivial and surprising because such a fundamental assumption is not at all assured to be true in the present nonequilibrium open optical system [34]. On the other hand, C stat of the NMFA with σ = 0.15 and 0.75 show flat specific heats, which may be due to the fact that mean-field approximation neglects some fluctuations (see the Appendix).…”

“…1 also shows (noise-less) mean-field results (see Appendix), exact results calculated by the Wang-Landau method [38], and experimental results from Ref. [34]. Note that, without spontaneous symmetry breaking, the root mean square magnetization has a finite value when the system favors the magnetic ordered phase, and thus, we use this quantity to investigate magnetization in the finite system.…”

“…This unphysical behavior is conspicuous for large σ = 1.25. We should note that the statistical and thermodynamic relation- ship C = β 2 ( H 2 − H 2 ) = T ∂S/∂T is not satisfied by the NMFA at all, while such a relationship is satisfied by the CIM at high temperature [34]. Note that the breakdown of the relationship in the CIM results at low temperature is not similar to the breakdown of the NMFA; the CIM captures the qualitative feature, the peak, of the specific heats.…”

“…On the other hand, the CIM captures the features of the exact results, with gradual and smooth changes in both U and S. Note that, in the CIM experiments, the inverse temperature β was derived from the information on U by using maximally likelihood method, so that the agreement between exact and CIM results is not surprising (see Ref. [34]). However, independently obtained quantity S also shows good agreement, suggesting that the CIM can capture the features of the exact solution.…”

“…Thus, the validity of the simple mean-field assumption is suspect if it is applied to the measurement feedback procedure in the CIM. In fact, our recent experiments show that the CIM can capture the phase transition of the two-dimensional Ising model beyond the capacity of the mean-field approximation [34]. Note that another kind of optical Ising machine has reproduced a mean-field description of the phase transition in spin-grass Ising models [16].…”