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

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

Figure 1

(a) The experimental setup of CIM [15, 16]. (b) A Simplified model of CIM.

Figure 2

(a) Time evolution of the expectation values of the in-phase components $〈X_1〉$ [green (light gray) line] and $〈X_2〉$ [blue (dark gray) line]. This single trajectories of $〈X_1〉$ and $〈X_2〉$ are generated by the ensemble average over 10 000 Brownian particles. (b) Magnified picture of Fig. 2 near the decision-making point. (c) Average photon numbers in the DOPO cavities. (d) Measured values of $X_1$ and $X_2$ by the homodyne detectors. (e) Variances ${\sigma }_1^2$ and ${\sigma }_2^2$ of the in-phase components. (f) Time evolution of the skewness of the in-phase components $〈\mathrm{\Delta }{X}^3〉$. (g) Time evolution of the expectation values of the in-phase components $〈X_1〉$ and $〈X_2〉$ under the Gaussian approximation. (h) Magnified view of (g) immediately above the threshold. (i) Variances of the in-phase components ${\sigma }_1^2$ and ${\sigma }_2^2$ under the Gaussian approximation.

Figure 3

Time evolution of the probabilities of (a) $P(x>0)$ and (b) $P(x<0)$ in DOPO1 whose final destination is $|\uparrow \downarrow 〉$ (or $〈\widehat{x}〉>0$). Blue solid lines show the probability calculated from Eq. (20), and green dotted lines show the probability calculated from the normal distribution with the same means and variances.

Figure 4

(a) Success rate vs connection strength $\zeta$ for the exact replicator dynamics model [blue (dark gray online)] and Gaussian approximation [orange (light gray online)]. (b) Success rate vs connection strength $\zeta$ for the measurement-feedback-based CIM [blue (dark gray online)] and optical delay line coupling CIM [orange (light gray online)]. (c) Success rate comparison; the same as (a) with two simulation methods in the case of $\kappa =0.01$ (d) Success rate comparison with two CIM models in the same condition with (c).

Figure 5

(a) Almost symmetric potential for the DOPO field and two wave packets with a small $\zeta$ value. (b) Asymmetric potential for the DOPO field with a large $\zeta$ value.