Quantum Projection Filter Model for the Coherent Ising Machine

A novel optoelectronic system, the coherent Ising machine (CIM), uses networks of optical parametric oscillators (OPOs) in conjunction with a measurement-based feedback scheme to represent and solve fully-connected Ising spin optimization problems. Solutions to these problems have wide-ranging applicability—data set analysis, circuit design, drug discovery and scheduling. Experimentally, the CIM has been demonstrated to successfully find exact or approximate solutions to a variety of hard optimization problems with up to 2,000 all-to-all connected spins. However, whether the CIM offers any computational advantage over conventional computers is an open question that requires understanding the (potentially quantum) dynamics that govern information processing in the CIM.

As a step towards characterizing CIM system dynamics and formally benchmarking any computational advantage afforded by these dynamics, we describe a model reduction approach that can facilitate scalable numerical studies of the CIM via dynamical simulation. Motivated by the low nonlinearity and high optical losses in the system, we consider the CIM dynamics to be described by a series of time evolutions of the system state on a Gaussian manifold of states, using Gaussian maps to represent the operations performed by beam splitters, measurement, and feedback injections. In particular, to ensure that the propagation of the OPO modes through the χ(2) waveguide is computationally tractable, we employ a quantum projection filter technique to differentially project the quantum dynamics induced by the optical nonlinearity onto the Gaussian manifold. In this work, we present the aforementioned model reduction and derive a set of filter-projected differential equations required for integrating the reduced model into numerical simulations of the CIM with nonlinear pump and signal dynamics. This approach may be useful for describing other photonic systems where nonlinear optical effects coexist with sufficiently strong linear losses.

We also provide a brief review of quantum optics and the Gaussian state formalism, along with a description of the measurement-based CIM and a summary of the important experimental results in the sub-field of CIM research.