Approximate optimization of MAXCUT with a local spin algorithm

TL;DR

This paper evaluates a local tensor algorithm for MAXCUT, demonstrating it often outperforms traditional solvers and gradient descent, and relates it to imaginary-time quantum dynamics.

Contribution

It introduces practical benchmarking of local tensor methods for MAXCUT, proposing hyperparameter formulas and showing superior performance over gradient descent and commercial solvers.

Findings

Local tensor method outperforms gradient descent on MAXCUT instances.

In some cases, it beats commercial solvers by up to two orders of magnitude.

The method closely follows discretized imaginary-time dynamics.

Abstract

Local tensor methods are a class of optimization algorithms that was introduced in [Hastings,arXiv:1905.07047v2][1] as a classical analogue of the quantum approximate optimization algorithm (QAOA). These algorithms treat the cost function as a Hamiltonian on spin degrees of freedom and simulate the relaxation of the system to a low energy configuration using local update rules on the spins. Whereas the emphasis in [1] was on theoretical worst-case analysis, we here investigate performance in practice through benchmarking experiments on instances of the MAXCUT problem.Through heuristic arguments we propose formulas for choosing the hyperparameters of the algorithm which are found to be in good agreement with the optimal choices determined from experiment. We observe that the local tensor method is closely related to gradient descent on a relaxation of maxcut to continuous variables, but consistently outperforms gradient descent in all instances tested. We find time to solution achieved by the local tensor method is highly uncorrelated with that achieved by a widely used commercial optimization package; on some MAXCUT instances the local tensor method beats the commercial solver in time to solution by up to two orders of magnitude and vice-versa. Finally, we argue that the local tensor method closely follows discretized, imaginary-time dynamics of the system under the problem Hamiltonian.