  Quantum Boltzmann machine extends the classical Boltzmann machine learning to
the quantum regime, which makes its power to simulate the quantum states beyond
the classical probability distributions. We develop the BFGS algorithm to study
the corresponding optimization problem in quantum Boltzmann machine, especially
focus on the target states being a family of states with parameters. As an
typical example, we study the target states being the real symmetric two-qubit
pure states, and we find two obvious features shown in the numerical results on
the minimal quantum relative entropy: First, the minimal quantum relative
entropy in the first and the third quadrants is zero; Second, the minimal
quantum relative entropy is symmetric with the axes $y=x$ and $y=-x$ even with
one qubit hidden layer. Then we theoretically prove these two features from the
geometric viewpoint and the symmetry analysis. Our studies show that the
traditional physical tools can be used to help us to understand some
interesting results from quantum Boltzmann machine learning.
