Evolved Quantum Boltzmann Machines

Abstract

We introduce evolved quantum Boltzmann machines as a variational ansatz for quantum optimization and learning tasks. Given two parameterized Hamiltonians $G(θ)$ and $H(ϕ)$, an evolved quantum Boltzmann machine consists of preparing a thermal state of the first Hamiltonian $G(θ)$ followed by unitary evolution according to the second Hamiltonian $H(ϕ)$. Alternatively, one can think of it as first realizing imaginary time evolution according to $G(θ)$ followed by real time evolution according to $H(ϕ)$. After defining this ansatz, we provide analytical expressions for the gradient vector and illustrate their application in ground-state energy estimation and generative modeling, showing how the gradient for these tasks can be estimated by means of quantum algorithms that involve classical sampling, Hamiltonian simulation, and the Hadamard test. We also establish analytical expressions for the Fisher-Bures, Wigner-Yanase, and Kubo-Mori information matrix elements of evolved quantum Boltzmann machines, as well as quantum algorithms for estimating each of them, which leads to at least three different general natural gradient descent algorithms based on this ansatz. Along the way, we establish a broad generalization of the main result of [Luo, Proc. Am. Math. Soc. 132, 885 (2004)], proving that the Fisher-Bures and Wigner-Yanase information matrices of general parameterized families of states differ by no more than a factor of two in the matrix (Loewner) order, making them essentially interchangeable for training when using natural gradient descent.