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arXiv:1903.01359

Realizing Quantum Boltzmann Machines Through Eigenstate Thermalization

Eric R. Anschuetz, Yudong Cao

Published Mar 4, 2019• 15 citations
arXivHTMLPDF

Abstract

Quantum Boltzmann machines are natural quantum generalizations of Boltzmann machines that are expected to be more expressive than their classical counterparts, as evidenced both numerically for small systems and asymptotically under various complexity theoretic assumptions. However, training quantum Boltzmann machines using gradient-based methods requires sampling observables in quantum thermal distributions, a problem that is NP-hard. In this work, we find that the locality of the gradient observables gives rise to an efficient sampling method based on the Eigenstate Thermalization Hypothesis, and thus through Hamiltonian simulation an efficient method for training quantum Boltzmann machines on near-term quantum devices. Furthermore, under realistic assumptions on the moments of the data distribution to be modeled, the distribution sampled using our algorithm is approximately the same as that of an ideal quantum Boltzmann machine. We demonstrate numerically that under the proposed training scheme, quantum Boltzmann machines capture multimodal Bernoulli distributions better than classical restricted Boltzmann machines with the same connectivity structure. We also provide numerical results on the robustness of our training scheme with respect to noise.

DOI
10.48550/arxiv.1903.01359

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Quantum Boltzmann machines are natural quantum generalizations of Boltzmann machines that are expected to be more expressive than their classical counterparts, as evidenced both numerically for small systems and asymptotically under various complexity theoretic assumptions. However, training quantum Boltzmann machines using gradient-based methods requires sampling observables in quantum thermal distributions, a problem that is NP-hard. In this work, we find that the locality of the gradient observables gives rise to an efficient sampling method based on the Eigenstate Thermalization Hypothesis, and thus through Hamiltonian simulation an efficient method for training quantum Boltzmann machines on near-term quantum devices. Furthermore, under realistic assumptions on the moments of the data distribution to be modeled, the distribution sampled using our algorithm is approximately the same as that of an ideal quantum Boltzmann machine. We demonstrate numerically that under the proposed training scheme, quantum Boltzmann machines capture multimodal Bernoulli distributions better than classical restricted Boltzmann machines with the same connectivity structure. We also provide numerical results on the robustness of our training scheme with respect to noise.
Quantum machine learning explores the devising and implementing of quantum software to enable the speedup for a classical machine learning counterpart, which has attracted extensive attention. The quantum Boltzmann machine is a promising quantum machine learning model and can be executed on near-term quantum devices to demonstrate quantum supremacy. We investigate an efficient algorithm for training a quantum Boltzmann machine based on a continuous-time quantum cluster Monte Carlo method realized with a dimension-expanded equivalent Hamiltonian. The proposed algorithm is capable of efficiently training various types of quantum Boltzmann machines, including a two-dimensional random transverse Ising Hamiltonian and other stochastic $\ensuremath{\kappa}$-local $(\ensuremath{\kappa}\ensuremath{\ge}2)$ Hamiltonians. We further apply a quantum mean-field method to speed up the training process through discarding the quantum fluctuations of visible-hidden interactions. The experiment results show that our algorithm-trained quantum Boltzmann machine is more efficient on a small and large data set in terms of network convergence and time complexity.
Abstract The quantum Boltzmann machine (QBM) is a generative machine learning model for both classical data and quantum states. Training the QBM consists of minimizing the relative entropy from the model to the target state. This requires QBM expectation values which are computationally intractable for large models in general. It is therefore important to develop heuristic training methods that work well in practice. In this work, we study a heuristic method characterized by a nested loop: the inner loop trains the β-variational quantum eigensolver (β-VQE) by Liu et al (2021 Mach. Learn.: Sci. Technol. 2 025011) to approximate the QBM expectation values; the outer loop trains the QBM to minimize the relative entropy to the target. We show that low-rank representations obtained by β-VQE provide an efficient way to learn low-rank target states, such as classical data and low-temperature quantum tomography. We test the method on both classical and quantum target data with numerical simulations of up to 10 qubits. For the cases considered here, the obtained QBMs can model the target to high fidelity. We implement a trained model on a physical quantum device. The approach offers a valuable route towards variationally training QBMs on near-term quantum devices.
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.
A Boltzmann machine is a powerful tool for modeling probability distributions that govern the training data. A thermal equilibrium state is typically used for the Boltzmann machine learning to obtain a suitable probability distribution. The Boltzmann machine learning consists of calculating the gradient of the loss function given in terms of the thermal average, which is the most time-consuming procedure. Here, we propose a method to implement the Boltzmann machine learning by using noisy intermediate-scale quantum devices. We prepare an initial pure state that contains all possible computational basis states with the same amplitude, and we apply a variational imaginary time simulation. Readout of the state after the evolution in the computational basis approximates the probability distribution of the thermal equilibrium state that is used for the Boltzmann machine learning. We perform the numerical simulations of our scheme and confirm that the Boltzmann machine learning works well. Our scheme leads to a significant step toward an efficient machine learning using quantum hardware.
Abstract Quantum Boltzmann machines (QBMs) are machine-learning models for both classical and quantum data. We give an operational definition of QBM learning in terms of the difference in expectation values between the model and target, taking into account the polynomial size of the data set. By using the relative entropy as a loss function, this problem can be solved without encountering barren plateaus. We prove that a solution can be obtained with stochastic gradient descent using at most a polynomial number of Gibbs states. We also prove that pre-training on a subset of the QBM parameters can only lower the sample complexity bounds. In particular, we give pre-training strategies based on mean-field, Gaussian Fermionic, and geometrically local Hamiltonians. We verify these models and our theoretical findings numerically on a quantum and a classical data set. Our results establish that QBMs are promising machine learning models.
The promise of quantum neural nets, which utilize quantum effects to model complex data sets, has made their development an aspirational goal for quantum machine learning and quantum computing in general. Here we provide methods of training quantum Boltzmann machines. Our work generalizes existing methods and provides additional approaches for training quantum neural networks that compare favorably to existing methods. We further demonstrate that quantum Boltzmann machines enable a form of partial quantum state tomography that further provides a generative model for the input quantum state. Classical Boltzmann machines are incapable of this. This verifies the long-conjectured connection between tomography and quantum machine learning. Finally, we prove that classical computers cannot simulate our training process in general unless BQP=BPP\text{BQP}=\text{BPP}BQP=BPP, provide lower bounds on the complexity of the training procedures and numerically investigate training for small nonstoquastic Hamiltonians.