Group theory on quantum Boltzmann machine

TL;DR

This paper introduces a group-theoretic framework to analyze symmetries in quantum Boltzmann machines, revealing how symmetries influence target states and solutions, and providing a systematic construction method.

Contribution

It develops a novel group theory approach for quantum Boltzmann machines, including a systematic construction procedure and a numerical verification algorithm.

Findings

Symmetry transformations relate equivalent target states.

Optimal solutions invariant under symmetry are equivalent.

A systematic method to construct symmetry groups for qubit-based models.

Abstract

Group theory is extremely successful in characterizing the symmetries in quantum systems, which greatly simplifies and unifies our treatments of quantum systems. Here we introduce the concept of the symmetry for a quantum Boltzmann machine and develop a group theory to describe the symmetry. This symmetry implies not only that all the target states related with the symmetry transformations are equivalent, but also that for a given target state all the optimal solutions related with the symmetry transformations that keeps the target state invariant are equivalent. For the Boltzmann machines built on qubits, we propose a systematic procedure to construct the group, and develop a numerical algorithm to verify the completeness of our construction.