The random energy landscape of soft-spin networks and its application to combinatorial optimizations

Building neuromorphic analog hardware to solve large-scale optimization problems has been a central challenge spanning physics and computer science. We analyze the computational performance of one such machine, a network of coupled optical parametric oscillators that act as soft-spins confined in a double-well potential. Their remarkable capacity for solving Ising encodable combinatorial optimization problems has been intensively analyzed in many optical experiments and computer simulations. However, we lack theoretical insights into why such networks can be efficient optimizers. We tackle this question by elucidating the geometry of the network's high dimensional energy landscape through the Kac-Rice formula with full replica symmetry and supersymmetry breaking. Our analysis reveals a hierarchy of geometric phase transitions in the landscape that facilitates optimization dynamics. In particular, we reveal a geometric phase where the non-convex landscape's local minima are almost all global minima, which allows the network to find solutions significantly better than naive spectral methods. We also find the existence of a phase where we cannot adiabatically follow the global minima, which creates a finite energy gap between the networks' solution and the optimal lowest energy.