Quantum model for coherent Ising machines: Discrete-time measurement feedback formulation

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.

Figure 1

(a) The experimental setup of the coherent Ising machine (CIM) using a degenerate optical parametric oscillator network and discrete-time measurement feedback processes. $N_{\mathrm{pulse}}$ pulses are arranged at regular intervals and running around the ring cavity. The amplitude of the feedback pulse injected to the pulse $j$ is calculated by the FPGA from the measurement results of all the other pulses. (b) The corresponding theoretical model. The first beam splitter, with transmittance rate $T^{\prime }$, represents the background loss along the ring cavity.

Figure 2

Contour maps of the density matrices $\langle x\left|\rho \right|x^{\prime }\rangle$ of typical quantum states as functions of $x+x^{\prime }$ and $x-x^{\prime }$. $\langle x\left|\rho \right|x^{\prime }\rangle$ are real functions in the cases of these states. (a) A vacuum state $|0\rangle$, (b) a one-photon state $|1\rangle$, (c) an antisqueezed vacuum state, (d) a thermal state ($\langle n\rangle =2$), (e) a cat state of coherent states $\frac{1}{\sqrt{2}}\left(\right|\alpha \rangle +|-\alpha \rangle )$, and (f) a classical ensemble of coherent states $\frac{1}{2}\left(\right|\alpha \rangle \langle \alpha |+|-\alpha \rangle \langle -\alpha \left|\right)$.

Figure 3

Time evolution of typical conditional density matrices of the two oscillators governed by the measurement results $x_m$ with $T=0.99$. Other parameters are shown in Table 1. (a) The evolution of $\langle x\rangle$ and $\langle \mathrm{\Delta }{x}^2\rangle$. The solid blue lines and dashed green lines represent the values of the signal pulse No. 1 and No. 2, respectively. (b) The contour maps of the typical conditional density matrices $<x\left|\rho \right|x^{\prime }>$ in front of the PSA plotted on the coordinates $x-x^{\prime }$ and $x+x^{\prime }$. The figures on the upper and lower row represent the density matrices of the signal pulse No. 1 and No. 2, respectively. In this system, all elements of the density matrices are real numbers.

Figure 4

The contour maps of typical conditional density matrices $<x\left|\rho \right|x^{\prime }>$ of two oscillators governed by the measurement results $x_m$ with (a) $T=0.999$, (b) $T=0.99$, (c) $T=0.9$, and (d) $T=0.5$ at round trips of $N=0,30,60,150$ in front of and behind the PSA. Other numerical parameters are shown in Table 1. In this system, all elements of the density matrices are real numbers. (e) The Wigner function $W(x+x^{\prime },p)=\frac{1}{\pi }{\int }_{-\infty }^{\infty }\langle x\left|\rho \right|x^{\prime }\rangle e^{2ip(x-x^{\prime })}d(x-x^{\prime })$ of the density matrices at $N=60$ in Fig. 4.

Figure 5

$\langle \mathrm{\Delta }{x}^2\rangle$ vs $\langle \mathrm{\Delta }{p}^2\rangle$ for a typical conditional density matrix of a signal pulse governed by the sequence of randomly determined measurement results $x_m$, with various measurement strengths. (a) The left cyan line and the right red line represent $\langle \mathrm{\Delta }{x}^2\rangle$ vs $\langle \mathrm{\Delta }{p}^2\rangle$ for $T=0.5$ and 0.9, respectively. (b) The four lines represent $\langle \mathrm{\Delta }{x}^2\rangle$ vs $\langle \mathrm{\Delta }{p}^2\rangle$ for $T=0.5,0.9,0.99$, and 0.999 in the order left to right. (c) The left blue line and the right green line represent $\langle \mathrm{\Delta }{x}^2\rangle$ vs $\langle \mathrm{\Delta }{p}^2\rangle$ in front of and behind the PSA, respectively, in the case of $T=0.5$. The dashed curves represent the line of the minimum uncertainty product $\langle \mathrm{\Delta }{x}^2\rangle \langle \mathrm{\Delta }{p}^2\rangle =1/4$. The initial state of the signal pulse is the vacuum state ($\langle \mathrm{\Delta }{x}^2\rangle =\langle \mathrm{\Delta }{p}^2\rangle =0.5$).

Figure 6

The probability of success $P$ for various values of background loss rate $1-T^{\prime }$ with three-pump schedule for net linear gain. (a) Slowly increasing pump rate, (b) rapidly increasing pump rate, and (c) constant pump rate. The definition of the success rate is ${\int }_{x_1{x}_2<0}\langle x_1|\langle x_2\left|\rho \right|x_1\rangle |x_2\rangle d{x}_{1}d{x}_2$, where $x_1$ and $x_2$ are the in-phase amplitudes for the two oscillators. These values are calculated from the averages of the probabilities of 3000 conditional density matrices. The red dotted, green dash-dot, blue dashed, and black solid lines represent the cases of $T^{\prime }=1.0,0.9,0.7,$ and 0.5, respectively. Other parameters for the numerical simulation are shown in Table 2.