Coherent Ising Machines on Photonic Integrated Circuits

Introduction

Ising machines offer high computational capabilities as next-generation hardware accelerators for solving combinatorial optimization problems. These problems are classified as NP-hard, meaning they cannot be solved in polynomial time on classical computers as the problem size increases. The Ising model, a fundamental concept in statistical mechanics, provides a versatile framework for representing and solving such problems.

The Ising model depicts an undirected graph with binary Ising spin nodes (xi = ±1) interconnected by Ising coupling interactions (Jij). The goal is to find spin configurations that minimize the Ising Hamiltonian, H = −(1/2)∑i≠jJijxixj, given a specific coupling interaction matrix. Solving an Ising problem is equivalent to solving other NP-hard problems like Maxcut and the traveling salesman problem.

Coherent Ising Machines (CIMs) solve Ising problems by implementing the Ising model and minimizing its Hamiltonian over time. Figure 1 illustrates the workflow of using an Ising machine as a Maxcut solver.

Fig. 1. Workflow of an Ising machine as a Maxcut solver: A Maxcut problem defined by an undirected graph is encoded into an Ising coupling matrix. This matrix is input into an Ising machine optimized for this problem. Through a noisy ground state search, the machine evolves to potentially solve the Maxcut problem, returning the solution as the ground-state Ising spin configuration.

Integrated CIM Model

Most CIMs are based on optical parametric oscillators (OPOs), requiring large experimental setups. As a compact alternative, integrated CIMs have been proposed for implementation on photonic integrated circuits (PICs). Figure 2 shows a schematic of an integrated CIM on a PIC.

Fig. 2. Schematic of a CIM on a photonic integrated circuit, featuring three main components: a nonlinear spin generator, an Ising coupling matrix multiplier, and feedback loops. The Ising machine operates in either subcritical or supercritical pitchfork regimes, with nonlinear spins and the matrix multiplier implemented using microring resonators and MZI mesh.

The time differential equation for each spin amplitude (xi) in the integrated CIM system can be approximated as a fifth-order polynomial transfer function:

Here, xi represents the ith light amplitude or analog spin amplitude, with the sign indicating the binary spin direction. r is the linear gain, η and ζ are the third-order and fifth-order coefficients, respectively. These parameters (r, η, and ζ) are tunable hyper-parameters that can be optimized for specific problems.

The fifth-order nonlinearity enables supercritical and subcritical operation regimes with hysteresis in the bifurcation diagram. Hysteresis is a phenomenon where a system experiences a time lag in response to external changes, acting as a "memory effect" that can enhance robustness against external random noise. By tuning the Ising machine hyper-parameters in a large noise regime, the hysteresis can be engineered to exploit the system's computational power.

Results

To demonstrate the computational capability and scalability of the fifth-order model, numerical experiments were performed on benchmark Maxcut problems from the BiqMac library. After optimizing the Ising machine hyper-parameters, the relative success rate (SR) results shown in Figure 3 illustrate that the fifth-order CIM model outperforms the state-of-the-art OPO CIM model in 23 out of 30 Maxcut instances, with an average of 60% increased SR.

Fig. 3. Relative success rate (SR) for Biqmac Maxcut instances with N = 60, 80, 100 nodes and 50% edge density, compared with simulation results of the state-of-the-art CIM in [6]. The SR refers to the probability of finding the ground state during one simulation trial, which was obtained by 100 trials per instance.

The advantages are more pronounced in larger instances with N = 80 and 100 nodes. This indicates that involving more tunable hyper-parameters and enabling hysteresis with large noise in the optimization process enhances the computational power of the Ising machine.

Conclusion

The simulations of coherent Ising machines (CIMs) on silicon photonics, having a fifth-order nonlinearity operated in the large-noise regime, show competitive scalability compared to other state-of-the-art CIMs. The fifth-order nonlinearity, originating from the nonlinearity of microring resonators on the photonic circuit, enables supercritical and subcritical operation regimes with hysteresis. By tuning the Ising machine hyper-parameters in a large noise regime, the hysteresis can be engineered to exploit the system's computational power. The numerical experiments on benchmark Maxcut problems demonstrate the advantages of the fifth-order CIM model, particularly for larger problem instances. This work paves the way for compact and scalable Ising machine implementations on photonic integrated circuits.

Reference

[1] R. Shi, T. Van Vaerenbergh, F. Böhm, and P. Bienstman, "Coherent Ising machines on photonic integrated circuits," in Proceedings of the IEEE International Conference on Photonics, Ghent, Belgium, 2024, pp. 1-2. doi: 979-8-3503-9404-7/24/$31.00.