Geometry and symmetry in quantum Boltzmann machine

TL;DR

This paper develops an optimization algorithm for quantum Boltzmann machines and reveals geometric and symmetry properties of target quantum states, enhancing understanding of quantum state simulation.

Contribution

The paper introduces a BFGS algorithm for quantum Boltzmann machine optimization and provides a geometric and symmetry-based analysis of target states.

Findings

Minimal quantum relative entropy is zero in certain quadrants.

Symmetry of minimal quantum relative entropy with respect to axes y=x and y=-x.

Geometric proof of observed symmetry properties.

Abstract

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.