Generative modeling using evolved quantum Boltzmann machines

TL;DR

This work provides a practical framework for Born-rule generative modeling using evolved quantum Boltzmann machines by leveraging the Donsker–Varadhan variational representation of relative entropy and the evolved quantum Boltzmann gradient estimator. It formulates a minimax objective with a neural-network feature map and develops four hybrid quantum–classical algorithms (extragradient, two-timescale descent-ascent, follow-the-ridge, HessianFR) for training, along with analytical gradient/Hessian expressions and norm bounds. The approach extends to alternative distinguishability measures such as Rényi relative quasi-entropies and includes a linear-feature-space variant to preserve concavity and improve convergence guarantees. This framework advances practical training of EQBMs for Born-rule sampling and provides a foundation for future convergence analysis and numerical validation.

Abstract

Born-rule generative modeling, a central task in quantum machine learning, seeks to learn probability distributions that can be efficiently sampled by measuring complex quantum states. One hope is for quantum models to efficiently capture probability distributions that are difficult to learn and simulate by classical means alone. Quantum Boltzmann machines were proposed about one decade ago for this purpose, yet efficient training methods have remained elusive. In this paper, I overcome this obstacle by proposing a practical solution that trains quantum Boltzmann machines for Born-rule generative modeling. Two key ingredients in the proposal are the Donsker-Varadhan variational representation of the classical relative entropy and the quantum Boltzmann gradient estimator of [Patel et al., arXiv:2410.12935]. I present the main result for a more general ansatz known as an evolved quantum Boltzmann machine [Minervini et al., arXiv:2501.03367], which combines parameterized real- and imaginary-time evolution. I also show how to extend the findings to other distinguishability measures beyond relative entropy. Finally, I present four different hybrid quantum-classical algorithms for the minimax optimization underlying training, and I discuss their theoretical convergence guarantees.

Figures (2)

• Figure 1: (a) Quantum circuit for estimating the first term $-\frac{1}{2}\left\langle \left\{ O,\Phi_{\theta}(G_{j})\right\} \right\rangle _{\rho_{\theta}}$ in \ref{['eq:gradient-VQE-obj']}. For each execution of this circuit, a value $t\in\mathbb{R}$ is chosen randomly from the high-peak probability density $p(t)$ in \ref{['eq:high-peak-tent-def']}. (b) Quantum circuit for estimating the quantity $-\frac{i}{2}\left\langle \left[O,\Psi_{\phi}(H_{k})\right]\right\rangle _{\omega_{\theta,\phi}}$ in \ref{['eq:time-evolve-grad']}. For each execution of this circuit, a value $t\in\mathbb{R}$ is chosen uniformly at random from the unit interval $\left[0,1\right]$. The symbol "Had" denotes the Hadamard gate, and the symbol $S$ denotes the phase gate $S\coloneqq100i$.
• Figure 2: Plot of the function in \ref{['eq:rel-ent-not-convex-example']}, demonstrating that it is not convex in $\mu$.

Theorems & Definitions (4)

• Proposition 3: Gradient bounds
• proof
• Proposition 4: Hessian bounds
• proof