Representing arbitrary ground states of toric code by a restricted Boltzmann machine

TL;DR

This paper investigates how well Restricted Boltzmann Machines can represent toric code ground states, introduces modifications for arbitrary states, and extends the approach from Z2 to Z_n toric codes.

Contribution

It analyzes the limitations of local RBMs for toric code ground states and proposes a modified model with non-local connections for arbitrary states, including generalization to Z_n codes.

Findings

Local RBMs have limited representability for toric code ground states.

Introducing non-local connections enables RBMs to represent arbitrary ground states.

The modified model is analytically solvable and performs efficiently in practice.

Abstract

We systematically analyze the representability of toric code ground states by Restricted Boltzmann Machine with only local connections between hidden and visible neurons. This analysis is pivotal for evaluating the model's capability to represent diverse ground states, thus enhancing our understanding of its strengths and weaknesses. Subsequently, we modify the Restricted Boltzmann Machine to accommodate arbitrary ground states by introducing essential non-local connections efficiently. The new model is not only analytically solvable but also demonstrates efficient and accurate performance when solved using machine learning techniques. Then we generalize our the model from $Z_2$ to $Z_n$ toric code and discuss future directions.