Quantum Thermodynamics of Quantum Boltzmann Machines

Quantum Boltzmann machines are the quantum extension of classical Boltzmann machines. The classical Boltzmann machine is a generative probabilistic model to find a Boltzmann distribution, given classical data. In Ref. [1], the learning process of the classical restricted Boltzmann machine was analyzed in the context of statistical mechanics. The entropy, free energy, work, and Jarzynski equality were investigated. Here we study quantum thermodynamics of the training process of the quantum restricted Boltzmann machine with the hidden and the visible units for data given by density matrices. Similar to the thermodynamic analysis of the classical restricted Boltzmann machine, we calculate how internal energy, free energy and entropy, and entanglement between visible and hidden layers of the quantum restricted Boltzmann machine change during the learning. Quantum Jarzynski equality is examined. Also, we discuss how counter-diabatic driving could accelerate the learning of classical and quantum Boltzmann machines.

[1] S. Oh, A. Baggag, and H. Nha, Entropy 22, 538 (2020).