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Given the fundamental importance of combinatorial optimization across manydiverse application domains, there has been widespread interest in thedevelopment of unconventional physical computing architectures that can deliverbetter solutions with lower resource costs. These architectures embed discreteoptimization problems into the annealed, analog evolution of nonlineardynamical systems. However, a theoretical understanding of their performanceremains elusive, unlike the cases of simulated or quantum annealing. We developsuch understanding for the coherent Ising machine (CIM), a network of opticalparametric oscillators that can be applied to any quadratic unconstrainedbinary optimization problem. Here we focus on how the CIM finds low-energysolutions of the Sherrington-Kirkpatrick spin glass. As the laser gain isannealed, the CIM interpolates between gradient descent on the soft-spin energylandscape, to optimization on coupled binary spins. By exploiting spin-glasstheory, we develop a detailed understanding of the evolving geometry of thehigh-dimensional CIM energy landscape as the laser gain increases, findingseveral phase transitions, from flat, to rough, to rigid. Additionally, wedevelop a cavity method that provides a precise geometric interpretation ofsupersymmetry breaking in terms of the response of a rough landscape tospecific perturbations. We confirm our theory with numerical experiments, andfind detailed information about critical points of the landscape. Our extensiveanalysis of phase transitions provides theoretically motivated optimalannealing schedules that can reliably find near-ground states. This analysisreveals geometric landscape annealing as a powerful optimization principle andsuggests many further avenues for exploring other optimization problems, aswell as other types of annealed dynamics, including chaotic, oscillatory orquantum dynamics.

Geometric landscape annealing as an optimization principle underlying the coherent Ising machine