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arXiv:2501.03367 · Physical Review A

Evolved quantum Boltzmann machines

Michele Minervini, Dhrumil Patel, Mark M. Wilde

Published Jan 21, 2026• 2 citations
arXivHTMLPDF

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.

DOI
10.1103/k2hw-r25g

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We introduce evolved quantum Boltzmann machines as a variational ansatz for quantum optimization and learning tasks. Given two parameterized Hamiltonians G(θ)G(θ)G(θ) and H(ϕ)H(ϕ)H(ϕ), an evolved quantum Boltzmann machine consists of preparing a thermal state of the first Hamiltonian G(θ)G(θ)G(θ) followed by unitary evolution according to the second Hamiltonian H(ϕ)H(ϕ)H(ϕ). Alternatively, one can think of it as first realizing imaginary time evolution according to G(θ)G(θ)G(θ) followed by real time evolution according to H(ϕ)H(ϕ)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.
Thermal states play a fundamental role in various areas of physics, and they are becoming increasingly important in quantum information science, with applications related to semidefinite programming, quantum Boltzmann machine learning, Hamiltonian learning, and the related task of estimating the parameters of a Hamiltonian. Here we establish formulas underlying the basic geometry of parameterized thermal states, and we delineate quantum algorithms for estimating the values of these formulas. More specifically, we establish formulas for the Fisher--Bures and Kubo--Mori information matrices of parameterized thermal states, and our quantum algorithms for estimating their matrix elements involve a combination of classical sampling, Hamiltonian simulation, and the Hadamard test. These results have applications in developing a natural gradient descent algorithm for quantum Boltzmann machine learning, which takes into account the geometry of thermal states, and in establishing fundamental limitations on the ability to estimate the parameters of a Hamiltonian, when given access to thermal-state samples. For the latter task, and for the special case of estimating a single parameter, we sketch an algorithm that realizes a measurement that is asymptotically optimal for the estimation task. We finally stress that the natural gradient descent algorithm developed here can be used for any machine learning problem that employs the quantum Boltzmann machine ansatz.
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.
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 ε\varepsilonε-approximate stationary point of the energy function optimized over parameter space, while using a number of parameterized-thermal-state samples that is polynomial in ε−1\varepsilon^{-1}ε−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.
Abstract This work presents a novel realization approach to quantum Boltzmann machines (QBMs). The preparation of the required Gibbs states, as well as the evaluation of the loss function’s analytic gradient, is based on variational quantum imaginary time evolution, a technique that is typically used for ground-state computation. In contrast to existing methods, this implementation facilitates near-term compatible QBM training with gradients of the actual loss function for arbitrary parameterized Hamiltonians which do not necessarily have to be fully visible but may also include hidden units. The variational Gibbs state approximation is demonstrated with numerical simulations and experiments run on real quantum hardware provided by IBM Quantum. Furthermore, we illustrate the application of this variational QBM approach to generative and discriminative learning tasks using numerical simulation.
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; I also 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 an independent derivation of a 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.
The preparation of quantum Gibbs states is a fundamental challenge in quantum computing, essential for applications ranging from modeling open quantum systems to quantum machine learning. Building on the Meta-Variational Quantum Eigensolver framework proposed by Cervera-Lierta et al. and a problem-driven ansatz design, we introduce two meta-learning algorithms: Meta-Variational Quantum Thermalizer (Meta-VQT) and Neural Network Meta-VQT (NN-Meta VQT) for efficient thermal state preparation of parametrized Hamiltonians on Noisy Intermediate-Scale Quantum (NISQ) devices. Meta-VQT utilizes a fully quantum ansatz, while NN Meta-VQT integrates a quantum--classical hybrid architecture. Both use collective optimization over a finite set of training parameter values to prepare approximate Gibbs states at held-out parameter values within the sampled parameter domain. We validate our methods on Transverse Field Ising Models (TFIM) with up to eight qubits and on a two-qubit Heisenberg model with all field terms. Within these finite-size benchmarks, the trained models accurately prepare thermal states at test points inside the training interval. For the six and eight qubit cases of TFIM, meta-learned parameters when combined with appropriately designed ansatz serve as warm-start initializations, significantly outperforming random initializations in the optimization tasks. Furthermore, a 3 qubit Kitaev ring example showcases our algorithm's effectiveness across finite-temperature crossover regimes. Finally, we apply our algorithms to train a Quantum Boltzmann Machine (QBM) on a 2-qubit Heisenberg model with all field terms, achieving enhanced training efficiency, improved Gibbs state accuracy, and a 30-fold runtime speedup over existing techniques such as variational quantum imaginary time (VarQITE)-based QBM highlighting the practicality of meta-algorithm-based QBMs.