Quantum Boltzmann machine learning of ground-state energies

TL;DR

The paper develops a Hybrid Quantum–Classical algorithm (QBM-GSE) for estimating ground-state energies using quantum Boltzmann machines with parameterized thermal states. It derives a tractable gradient formula and proves gradient/Lipschitz properties, enabling an unbiased quantum gradient estimator (QBGE) built from Hadamard-test circuits, Hamiltonian simulation, and classical sampling. The main result is a convergence guarantee to an $\varepsilon$-stationary point with polynomial sample complexity in $\varepsilon^{-1}$, the number of parameters $J$, and $\|\alpha\|_1$, with explicit bounds when the Hamiltonian is local. This work positions QBMs as a promising alternative to VQE for ground-state energy estimation, potentially avoiding barren plateaus in certain regimes, and suggests broader applicability to optimization and learning tasks in quantum settings.

Abstract

Estimating the ground-state energy of Hamiltonians is a fundamental task for which it is believed that quantum computers can be helpful. Several approaches have been proposed toward this goal, including algorithms based on quantum phase estimation and hybrid quantum-classical optimizers involving parameterized quantum circuits, the latter falling under the umbrella of the variational quantum eigensolver. Here, we analyze the performance of quantum Boltzmann machines for this task, which is a less explored ansatz based on parameterized thermal states and which is not known to suffer from the barren-plateau problem. We delineate a hybrid quantum-classical algorithm for this task and rigorously prove that it converges to an $\varepsilon$-approximate stationary point of the energy function optimized over parameter space, while using a number of parameterized-thermal-state samples that is polynomial in $\varepsilon^{-1}$, the number of parameters, and the norm of the Hamiltonian being optimized. Our algorithm estimates the gradient of the energy function efficiently by means of a quantum circuit construction that combines classical random sampling, Hamiltonian simulation, and the Hadamard test. Additionally, supporting our main claims are calculations of the gradient and Hessian of the energy function, as well as an upper bound on the matrix elements of the latter that is used in the convergence analysis.

Figures (5)

• Figure 1: Quantum circuit that plays a role in realizing an unbiased estimate of $-\frac{1}{2}\left\langle \left\{ H,\Phi_{\theta}(G_{j})\right\}\right\rangle$. The Boltzmann gradient estimator combines this estimate with an unbiased estimate of $\left\langle H\right\rangle \left\langle G_{j}\right\rangle$, to realize an unbiased estimate of the gradient $\nabla_\theta f(\theta)$ in \ref{['eq-mt:gradient-alt']}.
• Figure 2: The non-convex landscape of the objective function given by \ref{['eq:example']}.
• Figure 3: The high-peak-tent probability density function $p(t)$, defined in \ref{['eq:high-peak-tent-density']}.
• Figure 4: Quantum primitive for estimating $\frac{1}{2}\operatorname{Tr}\!\left[\left(U_1^\dag U_0 + U_0^{\dagger}U_1\right)\rho\right]$. Note that the "Had" gate denotes the Hadamard gate.
• Figure 5: Quantum circuit corresponding to the $n^{\operatorname{th}}$ iteration of Algorithm \ref{['algo:est_first_term']}.

Theorems & Definitions (36)

• Theorem 1
• Definition 2: Lipschitz Continuity
• Definition 3: Smoothness
• Lemma 4: Lipschitz Constant for a Multivariate Function
• proof
• Lemma 5: Lipschitz Constant for a Multivariate Vector-Valued Function
• proof
• Definition 6: $\varepsilon$-Stationary Point
• Lemma 7: Hoeffding’s Inequality
• Lemma 8: SGD Convergence