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Generative training of quantum Boltzmann machines with hidden units

Nathan Wiebe, Leonard Wossnig

TL;DR

This work tackles the challenge of fully quantum generative training for quantum Boltzmann machines that include hidden units by formulating the training objective as quantum relative entropy and proposing two complementary gradient-solutions. The first method leverages a variational upper bound for a restricted class of QBMs with commuting hidden Hamiltonians, yielding tractable gradient expressions and practical efficiency. The second method develops a general gradient estimation framework that uses Fourier-like log approximations and high-order divided differences to approximate the exact gradient, supported by quantum subroutines such as Gibbs-state preparation, amplitude estimation, and the Hadamard test. Together, these approaches establish conditions under which QBMs can be trained on quantum devices and provide complexity analyses, while also outlining open problems in optimal Gibbs-state preparation and potential lower bounds. This advances practical quantum generative modeling by enabling training with hidden units and nontrivial quantum correlations.

Abstract

In this article we provide a method for fully quantum generative training of quantum Boltzmann machines with both visible and hidden units while using quantum relative entropy as an objective. This is significant because prior methods were not able to do so due to mathematical challenges posed by the gradient evaluation. We present two novel methods for solving this problem. The first proposal addresses it, for a class of restricted quantum Boltzmann machines with mutually commuting Hamiltonians on the hidden units, by using a variational upper bound on the quantum relative entropy. The second one uses high-order divided difference methods and linear-combinations of unitaries to approximate the exact gradient of the relative entropy for a generic quantum Boltzmann machine. Both methods are efficient under the assumption that Gibbs state preparation is efficient and that the Hamiltonian are given by a sparse row-computable matrix.

Generative training of quantum Boltzmann machines with hidden units

TL;DR

This work tackles the challenge of fully quantum generative training for quantum Boltzmann machines that include hidden units by formulating the training objective as quantum relative entropy and proposing two complementary gradient-solutions. The first method leverages a variational upper bound for a restricted class of QBMs with commuting hidden Hamiltonians, yielding tractable gradient expressions and practical efficiency. The second method develops a general gradient estimation framework that uses Fourier-like log approximations and high-order divided differences to approximate the exact gradient, supported by quantum subroutines such as Gibbs-state preparation, amplitude estimation, and the Hadamard test. Together, these approaches establish conditions under which QBMs can be trained on quantum devices and provide complexity analyses, while also outlining open problems in optimal Gibbs-state preparation and potential lower bounds. This advances practical quantum generative modeling by enabling training with hidden units and nontrivial quantum correlations.

Abstract

In this article we provide a method for fully quantum generative training of quantum Boltzmann machines with both visible and hidden units while using quantum relative entropy as an objective. This is significant because prior methods were not able to do so due to mathematical challenges posed by the gradient evaluation. We present two novel methods for solving this problem. The first proposal addresses it, for a class of restricted quantum Boltzmann machines with mutually commuting Hamiltonians on the hidden units, by using a variational upper bound on the quantum relative entropy. The second one uses high-order divided difference methods and linear-combinations of unitaries to approximate the exact gradient of the relative entropy for a generic quantum Boltzmann machine. Both methods are efficient under the assumption that Gibbs state preparation is efficient and that the Hamiltonian are given by a sparse row-computable matrix.

Paper Structure

This paper contains 16 sections, 15 theorems, 145 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution:
  3. Training Boltzmann Machines
  4. Variational training for restricted Hamiltonians
  5. Gradient based training for general Hamiltonians
  6. Conclusion
  7. Supplemental material
  8. Mathematical preliminaries
  9. Amplitude estimation
  10. The Hadamard test
  11. Deferred proofs
  12. Derivation of the variational bound
  13. Gradient estimation
  14. Operationalizing the gradient based training
  15. Approach 2: Training With Higher Order Divided Differences And Function Approximations
  16. ...and 1 more sections

Key Result

Lemma 2

Assume that the Hamiltonian $H$ of the quantum Boltzmann machine takes the form described in eq. eq:ham_form, where $\theta_k$ are the parameters which determine the interaction strength and $v_k,h_k$ are unitary operators. Furthermore, let $h_k \ket{h}=E_{h,k}\ket{h}$ be the eigenvalues of the hidd where $\alpha_h = \frac{e^{- \mathrm{Tr} \left[\rho \widetilde{H}_h \right]}}{\sum_h e^{- \mathrm{

Theorems & Definitions (30)

  • Definition 1
  • Lemma 2
  • Lemma 3
  • Theorem 4
  • Theorem 5
  • Definition 6: Operator monoticity
  • Claim 7
  • proof
  • Lemma 8: Von Neumann Trace Inequality
  • Lemma 9: Derivative of matrix logarithm haber2018notes
  • ...and 20 more