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Quantum Boltzmann Machine

Mohammad H. Amin, Evgeny Andriyash, Jason Rolfe, Bohdan Kulchytskyy, Roger Melko

arXiv:1601.02036v1quant-ph

TL;DR

Classical Boltzmann Machines do not directly provide an efficient gradient-estimation route for a transverse-field quantum model because noncommuting operators complicate likelihood derivatives. The paper introduces a bound-based QBM with sampling-estimable gradients, demonstrates small-size training against classical BM training, and cautions that QBM behavior does not always match BM behavior.

  • Problem

    Noncommuting operators in a transverse-field QBM make likelihood gradients nontrivial to estimate by sampling.

  • Method

    The paper introduces a lower bound on quantum probabilities, yielding an upper-bound objective whose gradients can be estimated by sampling.

  • Results

    Small-size examples show QBM learning the data distribution, with bound-based QBM serving as a proxy, and QBM outperforming BM in a restricted model.

  • Takeaways & Limitations

    Quantum mechanics is exploited in both the probabilistic model and its training, but the model's machine-learning potential remains unexplored.

  • Takeaways & Limitations

    QBM cannot generally sample conditional distributions by clamping as classical BM does, so replacing BM techniques requires careful examination.

Abstract

from arXiv · show

Inspired by the success of Boltzmann Machines based on classical Boltzmann distribution, we propose a new machine learning approach based on quantum Boltzmann distribution of a transverse-field Ising Hamiltonian. Due to the non-commutative nature of quantum mechanics, the training process of the Quantum Boltzmann Machine (QBM) can become nontrivial. We circumvent the problem by introducing bounds on the quantum probabilities. This allows us to train the QBM efficiently by sampling. We show examples of QBM training with and without the bound, using exact diagonalization, and compare the results with classical Boltzmann training. We also discuss the possibility of using quantum annealing processors like D-Wave for QBM training and application.

I. INTRODUCTION

The paper extends classical probabilistic modeling with a quantum Boltzmann Machine whose quantum mechanics is part of both the model and training process. It formulates learning as fitting visible-variable probabilities to the data distribution through likelihood optimization and sampling-based gradient estimates.

  • I. INTRODUCTION: A successful model learns a visible distribution resembling the data distribution closely enough to make predictions about unseen situations.The paper notes that such probabilistic models can support classification, clustering, filtering, compression, denoising, and inpainting.
  • I. INTRODUCTION: Quantum machine-learning proposals had generally used quantum processors to accelerate classical models, whereas this work makes the probabilistic model itself quantum.The proposed QBM is based on the Boltzmann distribution of a quantum Hamiltonian, exploiting quantum mechanics in both modeling and training.
  • I. INTRODUCTION: Classical Boltzmann Machines model binary visible and hidden units with a quadratic energy function and underpin deep belief networks and deep Boltzmann machines.The paper uses z_a ∈ {−1,+1} and distinguishes visible variables v from hidden variables h.
  • I. INTRODUCTION: Training tunes Hamiltonian parameters so the visible marginal distribution P_v approaches the data distribution P_data_v by maximizing average log-likelihood.The parameters include biases b_a and couplings w_ab.
  • I. INTRODUCTION: Gradient updates use differences between clamped and unclamped Boltzmann averages, commonly called the positive and negative phases.Clamping fixes visible variables to data, and the averages can be estimated by sampling when sampling is efficient.

II. QUANTUM BOLTZMANN MACHINE

The QBM replaces classical spins with qubits in a transverse-field Ising Hamiltonian, producing a generally non-diagonal density matrix and quantum Boltzmann probabilities. Noncommuting operators make exact likelihood gradients difficult to estimate by sampling, motivating a bound-based alternative.

  • II. QUANTUM BOLTZMANN MACHINE: The QBM uses a quantum Boltzmann density matrix whose visible marginal P_v gives the probability of each measured visible output.The visible marginal is obtained by tracing over hidden variables.
  • II. QUANTUM BOLTZMANN MACHINE: Adding transverse-field spin components makes the Hamiltonian non-diagonal, so its eigenstates become superpositions of classical visible-hidden states.Measurements in the σ_z basis yield classical ±1 outcomes.
  • II. QUANTUM BOLTZMANN MACHINE: QBM training minimizes negative log-likelihood, but noncommutation means ∂_θe^−H ≠ −e^−H∂_θH, preventing the classical expectation-based gradient derivation.The paper introduces imaginary-time expansions to analyze the noncommuting exponential derivative.
  • II. QUANTUM BOLTZMANN MACHINE: One gradient term can be estimated by sampling, whereas another cannot, making exact QBM training inefficient and impractical for large systems.This limitation motivates introducing an upper bound on negative log-likelihood, called bound-based QBM.

A. Bound-based QBM

Bound-based QBM replaces the intractable exact objective with an upper bound whose gradients can be estimated from clamped and unclamped sampling. However, this bound cannot learn transverse fields on visible variables and can be costly for large datasets.

  • A. Bound-based QBM: The Golden-Thompson inequality provides a lower bound for quantum probabilities, which induces an upper bound on the negative log-likelihood.The bound-based objective is denoted ˜L and is minimized instead of the exact loss L.
  • A. Bound-based QBM: Bound-based gradients retain differences between unclamped and clamped averages, estimated by sampling from Boltzmann distributions defined by H and H_v.The paper compares training with the exact objective and its upper bound using these sampled averages.
  • A. Bound-based QBM: The upper bound cannot learn the transverse field for visible variables: it drives Γ_ν toward zero, an artifact inconsistent with exact-gradient training.The transverse field may still be trained with the exact log-likelihood, but this rapidly becomes inefficient as the QBM grows.

B. Restricted QBM

Restricted QBM removes lateral connectivity among hidden qubits so the positive-phase calculations can be performed exactly. In the Γi → 0 limit, its expression reduces to the classical RBM result.

  • Restricted architecture: Restricting the hidden layer to no lateral connectivity makes hidden qubits uncoupled in the positive phase, enabling exact calculation of the relevant averages.The restriction is applied to the QBM architecture shown in Fig. 1b.
  • Classical limit: The restricted quantum expression reduces to the classical RBM expression when Γi → 0.The paper emphasizes that the quantum expression, rather than only its classical limit, is used for RQBM training.

III. SUPERVISED LEARNING

Supervised learning infers output labels from inputs by learning conditional or joint probability distributions. The paper distinguishes discriminative learning, which targets the conditional distribution, from generative learning, which models the joint distribution.

  • Supervised learning: Supervised learning infers a function from feature vectors x to labels y using a set of labeled examples (xi, yi).In probabilistic approaches, the most probable output conditional on x is selected as the label.
  • Learning objective: The training objective is to make the learned conditional or joint distribution close to the corresponding data distribution.For supervised learning, the desired conditional target is P_data_y|x; generative learning instead targets P_data_x,y.
  • Discriminative learning: Discriminative learning directly models the conditional distribution P_data_y|x for each input x.Its loss function is based on matching the data conditional distribution.
  • Generative learning: Generative learning models the joint distribution of inputs and outputs without separating input from output.Its loss includes a cross-entropy term measuring the difference between the training-input distribution and the model marginal distribution.

A. Generative learning

Generative QBM learning treats inputs and outputs jointly as visible data and trains a model whose joint distribution approaches the data distribution. However, quantum noncommutativity makes clamping an input an unreliable substitute for the true conditional distribution.

  • Generative learning: Generative QBM learning treats the input-output pair (x, y) jointly as visible data and trains its joint probability P_x,y toward the data distribution.After training, the QBM provides samples from the learned joint distribution.
  • Sampling challenge: When P_x is very small, reliably estimating P_y|x from samples of P_x,y and P_x requires a large number of samples.Rare inputs therefore create a sampling burden for the generative approach.
  • Quantum clamping: For classical Hamiltonians, clamping x permits efficient conditional sampling, but for quantum Hamiltonians the clamped distribution need not equal P_y|x.The difference follows when H does not commute with the input-clamping operator.
  • Quantum clamping: The approximation P_clamped_y|x ≈ P_y|x is not generally valid.The paper states that an example in Section IV-C demonstrates this failure of the approximation.

B. Discriminative learning

Discriminative QBM learning clamps the input in both training phases and represents the input through biases rather than input qubits. The positive phase can be efficient with restricted hidden connectivity, but the negative phase remains computationally costly.

  • Discriminative setup: Discriminative learning models the conditional output distribution by clamping the input x during both positive and negative phases.Because the input remains clamped, it acts as a bias on the other variables.
  • Input representation: The discriminative QBM does not need input qubits because fixed inputs only apply biases to the remaining variables.The Hamiltonian is conditioned on the particular input state x.
  • Input representation: Input features xμ need not be binary, allowing greater flexibility in supervised learning.The features enter through tunable input-to-qubit weights w_aμ.
  • Objective and optimization: The probability of measuring output y given input x is defined from the input-conditioned Hamiltonian H_x, with the negative log-likelihood as the loss.Training introduces an upper bound on the loss and computes its parameter derivatives for gradient descent.
  • Computational cost: Using restricted hidden connectivity keeps the positive phase efficient, whereas the negative phase may make gradient calculation expensive for large data sets.The negative phase requires a more elaborate sampling method unless very fast sampling is available.

IV. EXAMPLES

The paper evaluates QBM training on artificial data generated as mixtures of factorized Bernoulli distributions, using KL divergence to measure how closely learned and data distributions agree.

  • IV. EXAMPLES: Training data are generated as mixtures of M factorized distributions, with each mode concentrated around a randomly selected binary center.Each mode assigns probability p to aligning a qubit with its center; the mode probabilities depend on Hamming distance.
  • IV. EXAMPLES: All examples use p = 0.9 and M = 8.
  • IV. EXAMPLES: Learning quality is measured with KL divergence, the non-negative difference between the learned and target distributions.KL divergence equals zero only when the two distributions are identical.

A. Fully visible model

In a fully visible, fully connected 10-qubit model, exact-gradient BFGS training compares BM, QBM, and bound-based QBM. QBM learns the artificial data better than BM, while bQBM approaches QBM’s result.

  • A. Fully visible model: The experiment uses a fully connected N = 10-qubit model and trains BM, QBM, and bQBM with exact log-likelihood expectations.The QBM transverse-field parameters are restricted to a common value Γ, and training uses BFGS.
  • A. Fully visible model: QBM learns the data noticeably better than BM, while bQBM approaches a value close to QBM.
  • A. Fully visible model: The training analysis tracks classical and quantum average energies alongside the optimization trajectories.
  • A. Fully visible model: QBM lowers Γ initially, then increases it and converges to Γ = 2.5 with KL ≈0.42, compared with BM’s KL ≈0.62.

B. Restricted QBM

Restricted and supervised experiments show that QBM and bQBM can outperform BM, but consistent quantum evaluation and appropriate conditional-probability treatment are essential. Quantum-annealer training is proposed under freeze-out and hardware-control conditions.

  • B. Restricted QBM: In a restricted model with 8 visible and 2 hidden units, QBM outperforms BM and bQBM provides a good proxy for learning the quantum distribution.
  • B. Restricted QBM: Using classical expectations for the bQBM positive phase makes performance worse than BM because the two gradient terms are evaluated inconsistently.
  • C. Generative supervised learning: In supervised learning with 8 inputs and 3 outputs, QBM achieves better joint and conditional-distribution KL divergence than BM.The experiment assigns 3-bit labels to the eight mixture modes.
  • C. Generative supervised learning: As the transverse field grows, the clamped distribution diverges from the true conditional distribution and reaches a KL divergence much worse than classical BM.
  • V. QBM WITH A QUANTUM ANNEALING PROCESSOR: A quantum annealer can train a QBM if its controllable freeze-out point allows the dimensionless QBM parameters Γ, b_a, and w_ab to be tuned.Controlled sampling is currently limited to specially designed problems with very slow dynamics; hardware modifications may broaden applicability.

VI. CONCLUSION

The paper develops and evaluates a quantum Boltzmann machine using a transverse-field Ising Hamiltonian, sampling-based bounds, and exact diagonalization. It reports better data-distribution learning than classical Boltzmann machines in small examples, while identifying scaling and hardware questions for future work.

  • Method: A lower bound on the log-likelihood makes QBM gradient estimation amenable to sampling despite the transverse field.The paper trains using both the log-likelihood and its lower bound with exact diagonalization.
  • Results: In small-size examples, the QBM learned the data distribution better than a classical Boltzmann machine.The paper does not establish whether this advantage persists for larger systems or improves generalization.
  • Contribution: The QBM incorporates quantum mechanics directly into its probabilistic model through a quantum Boltzmann distribution, rather than using quantum hardware only to facilitate classical training.This distinguishes the approach from earlier quantum machine learning proposals and leaves the model’s machine learning potential unexplored.
  • Limitations: QBM training cannot generally reuse classical BM procedures: conditional distributions cannot be sampled by clamping, and other model differences may exist.The authors therefore call for careful examination before replacing BM with QBM techniques.
  • Future direction: Quantum annealers could support QBM training if hardware such as D-Wave were modified to provide quantum Boltzmann samples.The paper presents this as a possibility for future quantum information processing and machine learning research.
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