  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.
