Taming a non-convex landscape with dynamical long-range order: memcomputing Ising benchmarks

TL;DR

This paper demonstrates that classical memcomputing machines with memory can efficiently solve non-convex optimization problems, like Ising benchmarks, by leveraging collective dynamics and long-range order, similar to quantum annealing advantages.

Contribution

The study shows that memcomputing machines can solve Ising benchmarks in polynomial time using collective dynamics, highlighting the role of memory and long-range order.

Findings

Memcomputing machines solve Ising benchmarks efficiently.

Long-range order emerges in memcomputing dynamics.

Avalanche phases relate to success probability.

Abstract

Recent work on quantum annealing has emphasized the role of collective behavior in solving optimization problems. By enabling transitions of clusters of variables, such solvers are able to navigate their state space and locate solutions more efficiently despite having only local connections between elements. However, collective behavior is not exclusive to quantum annealers, and classical solvers that display collective dynamics should also possess an advantage in navigating a non-convex landscape. Here, we give evidence that a benchmark derived from quantum annealing studies is solvable in polynomial time using digital memcomputing machines, which utilize a collection of dynamical components with memory to represent the structure of the underlying optimization problem. To illustrate the role of memory and clarify the structure of these solvers we propose a simple model of these machines that demonstrates the emergence of long-range order. This model, when applied to finding the ground state of the Ising frustrated-loop benchmarks, undergoes a transient phase of avalanches which can span the entire lattice and demonstrates a connection between long-range behavior and their probability of success. These results establish the advantages of computational approaches based on collective dynamics of continuous dynamical systems.