Quest for quantum advantage: Monte Carlo wave-function simulations of the Coherent Ising Machine

TL;DR

The paper investigates whether quantum advantages can be realized in the Coherent Ising Machine (CIM) by performing fully quantum Monte Carlo Wave-Function simulations of small DOPO networks in highly quantum regimes. It models the CIM with a Hamiltonian plus master equation that include one- and two-photon dissipation and coherent couplings, and applies MCWF to simulate large Hilbert spaces across different initial states and dynamical strategies. Success probabilities are extracted from joint x-quadrature measurements, and purity is tracked to quantify decoherence, revealing that nonclassical initial states and time-dependent couplings can yield faster convergence in certain regimes. The findings suggest a potential quantum advantage in tailored, dynamically controlled CIM implementations, while also highlighting decoherence and scalability as key challenges requiring further investigation.

Abstract

The Coherent Ising Machine (CIM) is a quantum network of optical parametric oscillators (OPOs) intended to find ground states of the Ising model. This is an NP-hard problem, related to several important minimization problems, including the max-cut graph problem, and many similar problems. In order to enhance its potential performance, we analyze the coherent coupling strategy for the CIM in a highly quantum regime. To explore this limit we employ accurate numerical simulations. Due to the inherent complexity of the system, the maximum network size is limited. While master equation methods can be used, their scalability diminishes rapidly for larger systems. Instead, we use Monte Carlo wave-function methods, which scale as the wave-function dimension, and use large numbers of samples. These simulations involve Hilbert spaces exceeding $10^{7}$ dimensions. To evaluate success probabilities, we use quadrature probabilities. We demonstrate the potential for quantum computational advantage through improved simulation times and success rates in a low-dissipation regime, by using quantum superpositions and time varying couplings to give enhanced quantum effects.

Figures (9)

• Figure 1: Ising spin diagram of $M=3$, $M=4$ and $M=5$ spin arrangements with nearest neighbor interactions only, with uniform interaction strength ($J_{ij}=-1$, where $i$ and $j$ are nearest neighbor modes), with the sign of the $J_{12}$ interaction flipped ($J_{12}=J_{21}=1$). .
• Figure 2: Success rate variation with varying photon cut-off $N$. Simulation parameters are set to $\lambda=2.4$ and $g=0.6$. Here the couplings $J_{ij}$ are as in the $M=3$ case in Fig (\ref{['fig:Diag']}), with $\psi_{vac}$ as the initial state. .
• Figure 3: Evolution of the success rate for the anti-ferromagnetic spin problem for three cases, $M=3$, $M=4$ and $M=5$ . Three initial states are compared: the vacuum state $\psi_{vac}$ (Black), the coherent superposition state $\psi_{sup}$ (Blue) and the $M$-partite entangled state $\psi_{ent}$ (red). Simulation parameters are set to $\lambda=2.4$ and $g=0.6$, with a total simulation time of $t=8$ and $N_{steps}=1200$ time steps, averaged over $10^{4}$ realizations. Here $J_{ij}$ values are given in\ref{['fig:Diag']}.
• Figure 4: The time evolution of the success probability (top figure) and the purity (bottom figure), for different initial states, the vacuum state and a cat product state. The coupling matrix is given by Eq. (\ref{['eq:coupling_matrix']}), with $J_{12}=0.295$. The single-photon decay rate is $\gamma=1$, the two-photon decay rate is $g=0.75$, and a pump amplitude is $\lambda=1.125$. The total dimensionless simulation time is $T=3$, with $10^{4}$ samples and $600$ time steps. A photon cutoff number of $5$ is chosen. The two lines in each case indicate the sampling error.
• Figure 5: (a) Investigating the impact of initial states on purity and success rate for $M=3$ case in Fig (\ref{['fig:Diag']}), with varying initial states in the short time scale. Top Plots: Purity and Bottom Plots: Corresponding Success rates. Here we consider two settings, Left Plots: with high--quantum noise setting ($g=0.6$, $\gamma=1$ and $J_{coef}=1$) and Right Plots: low-quantum noise setting ($g=0.1$, $\gamma(t)=t/t_{max}$ and $J_{coef}(t)=3t/t_{max}+1$), with simulation time of $t_{max}=0.4$ with $t_{steps}=50$ using $10^{3}$ trajectories.