Extremizing a quadratic form can be computationally straightforward or difficult depending on the feasible domain over which variables are optimized. For example, maximizing E = xTVx for a real-symmetric matrix 𝑉 with 𝑥 constrained to a unit ball in 𝑅𝑁 can be performed simply by finding the maximum (principal) eigenvector of 𝑉, but can become computationally intractable if the domain of 𝑥 is limited to corners of the ±1 hypercube in 𝑅𝑁 (i.e., 𝑥 is constrained to be a binary vector). Many gain-loss physical systems, such as coherently coupled arrays of lasers or optical parametric oscillators, naturally solve minimum/maximum eigenvector problems (of a matrix of coupling coefficients) in their equilibration dynamics. In this paper we discuss recent case studies on the use of added nonlinear dynamics and real-time feedback to enforce constraints in such systems, making them potentially useful for solving difficult optimization problems. We consider examples in both classical and quantum regimes of operation.
Coherent Ising Machines (CIMs) are an emerging class of computational architectures that embed hard combinatorial optimization problems in the continuous dynamics of a physical system with analog degrees of freedom. While crisp theoretical results on the ultimate performance and scaling of such architectures are lacking, large-scale experimental prototypes have begun to exhibit promising results in practice. Our team at Stanford has begun to study the fundamental properties of CIM dynamics using a combination of techniques from statistical physics, random matrices, and dynamical systems theory. Many connections to recent work in neuroscience and deep learning are noted. Our work focuses specifically on CIMs that utilize the nonlinear threshold behavior of optical parametric oscillators to effect a soft (potentially glassy) transition between linear and binary dynamical regimes.
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