Fundamentals of quantum Boltzmann machine learning with visible and hidden units

TL;DR

This work establishes a quantum-framework for training QBMs with both visible and hidden units by deriving analytical gradient expressions for the quantum relative entropy between a target state and a QBM's visible marginal, incorporating modular-flow-based rotations to handle noncommutativity. It provides quantum algorithms to estimate these gradients via block-encodings, QSVT, and modular-flow subroutines, enabling hybrid quantum–classical optimization. The results cover fully quantum, quantum–classical, and classical–quantum QBMs, including restricted variants, and extend to Petz–Tsallis relative entropies with a novel matrix-power derivative. Collectively, the paper advances the tractability of quantum state learning and generative modeling with QBMs, offering concrete gradient formulas and estimation procedures with detailed complexity analyses and practical special cases.

Abstract

One of the primary applications of classical Boltzmann machines is generative modeling, wherein the goal is to tune the parameters of a model distribution so that it closely approximates a target distribution. Training relies on estimating the gradient of the relative entropy between the target and model distributions, a task that is well understood when the classical Boltzmann machine has both visible and hidden units. For some years now, it has been an obstacle to generalize this finding to quantum state learning with quantum Boltzmann machines that have both visible and hidden units. In this paper, I derive an analytical expression for the gradient of the quantum relative entropy between a target quantum state and the reduced state of the visible units of a quantum Boltzmann machine. Crucially, this expression is amenable to estimation on a quantum computer, as it involves modular-flow-generated unitary rotations reminiscent of those appearing in my prior work on rotated Petz recovery maps. This leads to a quantum algorithm for gradient estimation in this setting. I then specialize the setting to quantum visible units and classical hidden units, and vice versa, and provide analytical expressions for the gradients, along with quantum algorithms for estimating them. Finally, I replace the quantum relative entropy objective function with the Petz-Tsallis relative entropy; here I develop an analytical expression for the gradient and sketch a quantum algorithm for estimating it, as an application of a novel formula for the derivative of the matrix power function, which also involves modular-flow-generated unitary rotations. Ultimately, this paper demarcates progress in training quantum Boltzmann machines with visible and hidden units for generative modeling and quantum state learning.

Figures (1)

• Figure 1: Depiction of a quantum circuit that estimates $\left\langle G_{j}\right\rangle _{\Sigma_{v\to vh}^{\theta}\!\left(\rho\right)}$, the first term in \ref{['eq:gradient-fully-QBM']}. The first part of the circuit prepares a block-encoding of $\sigma_{v}^{-1/2}\sigma_{v}^{-is/2}$, which acts on the target state $\rho$. The last part of the circuit performs a swap test and measures the observable $e^{iG(\theta)t}G_{j}e^{-iG(\theta)t}$. In each execution of the circuit, the value $s$ is sampled from the logistic probability density $\beta(t)$ in \ref{['eq:logistic-prob-dens']}, and the value $t$ is sampled from the high-peak tent probability density $\gamma(t)$ in \ref{['eq:high-peak-tent-def']}.

Theorems & Definitions (42)

• Lemma 1: Derivative of matrix exponential
• proof
• Remark 2: Derivative of a thermal state
• Lemma 3: Derivative of matrix logarithm
• proof
• Lemma 4: Derivative of matrix power
• proof
• Remark 5
• Remark 6: Fréchet derivatives
• Remark 7