Effect of Coupling Discretization on Coherent-Ising-Machine-Implemented Hopfield Model

The coherent Ising machine (CIM) is being developed for solving combinatorial optimization problems scalably at high speed. The actual equipment of the measurement-feedback type of CIM (MFB-CIM) uses a field-programmable gate array (FPGA) to calculate the local field. Because the product of the coupling strength matrix and the measured amplitude vector of the optical parametric oscillator pulses must be calculated in real-time to achieve feedback, the coupling strength must have a low-bit discrete representation in the FPGA. By reducing the number of bits of coupling strength, the amount of memory needed to store the coupling matrix can be reduced and the calculation speed increased so that the MFB-CIM can implement more spins. Therefore, it is necessary to reduce the number of bits of coupling strength as much as possible while maintaining the performance in solving combinatorial optimization problems. In this paper, to evaluate the effect of discretization of the coupling strength in the MFB-CIM, we quantitatively evaluate the performance of the Hopfield model with discrete coupling when it is implemented in the CIM. The Hopfield model, which is a canonical model of Ising computation in information statistical mechanics, is defined in a coupling manner called the correlation-type interaction, in common with other models such as Sourlas code, a code division multiple access multiuser detector, and L0 regularization-based compressed sensing. We derived the macroscopic equations (MEs) of the Hopfield model with discrete coupling when it is implemented in the CIM by using the non-equilibrium statistical mechanics method we previously proposed. Moreover, we clarified the relationship between the number of bits of the coupling strength and the critical memory capacity for various pump rates.