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Quantum Boltzmann machine extends the classical Boltzmann machine learning tothe quantum regime, which makes its power to simulate the quantum states beyondthe classical probability distributions. We develop the BFGS algorithm to studythe corresponding optimization problem in quantum Boltzmann machine, especiallyfocus on the target states being a family of states with parameters. As antypical example, we study the target states being the real symmetric two-qubitpure states, and we find two obvious features shown in the numerical results onthe minimal quantum relative entropy: First, the minimal quantum relativeentropy in the first and the third quadrants is zero; Second, the minimalquantum relative entropy is symmetric with the axes $y=x$ and $y=-x$ even withone qubit hidden layer. Then we theoretically prove these two features from thegeometric viewpoint and the symmetry analysis. Our studies show that thetraditional physical tools can be used to help us to understand someinteresting results from quantum Boltzmann machine learning.

Geometry and symmetry in quantum Boltzmann machine