Group Theory on Quantum Boltzmann Machine

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 keep 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. (C) 2021 Elsevier B.V. All rights reserved.