Maximum-Entropy Inference with a Programmable Annealer

TL;DR

This paper demonstrates that a programmable Josephson junction array can perform maximum entropy inference, which can outperform maximum likelihood methods in certain noisy decoding tasks, and introduces a method to verify equilibrium sampling.

Contribution

It experimentally shows maximum entropy decoding using a quantum annealer and introduces a microscopic analytical method to confirm Boltzmann-like sampling.

Findings

Maximum entropy decoding can outperform maximum likelihood in noisy environments.

The annealer samples from a Boltzmann-like distribution.

Control errors, not thermalization, limit performance.

Abstract

Optimisation problems in science and engineering typically involve finding the ground state (i.e. the minimum energy configuration) of a cost function with respect to many variables. If the variables are corrupted by noise then this approach maximises the likelihood that the solution found is correct. An alternative approach is to make use of prior statistical information about the noise in conjunction with Bayes's theorem. The maximum entropy solution to the problem then takes the form of a Boltzmann distribution over the ground and excited states of the cost function. Here we use a programmable Josephson junction array for the information decoding problem which we simulate as a random Ising model in a field. We show experimentally that maximum entropy decoding at finite temperature can in certain cases give competitive and even slightly better bit-error-rates than the maximum likelihood approach at zero temperature, confirming that useful information can be extracted from the excited states of the annealing device. Furthermore we introduce a microscopic bit-by-bit analytical method which is agnostic to the specific application and use it to show that the annealing device samples from a highly Boltzmann-like distribution. Machines of this kind are therefore candidates for use in a wide variety of machine learning applications which exploit maximum entropy inference, including natural language processing and image recognition. We further show that the limiting factor for performance in our experiments is likely to be control errors rather than failure to reach equilibrium. Our work also provides a method for determining if a system is in equilibrium which can be easily generalized. We discuss possible applications of this method to spin glasses and probing the performance of the quantum annealing algorithm.