Diabatic quantum annealing for training energy-based generative models

1. Problem Statement

Training Energy-Based Models (EBMs), such as Restricted Boltzmann Machines (RBMs), requires generating unbiased samples from the Boltzmann distribution defined by the model's energy function.

• The Bottleneck: Classical training relies on Markov Chain Monte Carlo (MCMC) methods (e.g., Contrastive Divergence or Persistent Contrastive Divergence). These methods suffer from slow convergence and produce highly correlated samples, making large-scale training computationally expensive and inefficient.
• The Quantum Challenge: While analog quantum computers (quantum annealers) have been proposed as hardware samplers, previous attempts failed to provide reliable temperature control. The effective sampling temperature of quantum hardware was often treated as an empirical fitting parameter, leading to a misalignment between the hardware's output distribution and the theoretical target distribution required for training. This lack of control compromised reproducibility and the distinction between genuine quantum dynamics and fitting artifacts.

2. Methodology

The authors propose a framework that combines Diabatic Quantum Annealing (DQA) with a systematic temperature calibration method to train RBMs.

A. Diabatic Quantum Annealing (DQA) for Sampling

Instead of waiting for the system to reach the ground state (adiabatic), the system is evolved rapidly (diabatically) under a time-dependent Hamiltonian:
$$H(t) = A(t)H_{mixing} + B(t)H_{problem}$$

• Theoretical Basis: Recent theory establishes an analytic relation between the annealing schedule ($A(t), B(t)$) and an effective inverse temperature ($\beta_{integral}$).
• Key Insight: By controlling the annealing time ($\tau$) and schedule, the effective temperature of the generated samples can be prescribed in advance rather than inferred post-hoc.
• Implementation: The authors use the D-Wave Advantage2 quantum annealer. They identified that a fast annealing time of 5 ns with the default schedule yields an effective inverse temperature of $\beta \approx 1.5$.

B. Temperature Calibration and Rescaling

The authors identified a systematic deviation where the actual hardware temperature ($\beta_{dwave}$) was significantly lower (higher $\beta$) than the theoretical prediction ($\beta_{unitary}$), fluctuating by a factor of 5–7.

• Correction Method: They propose an analytic rescaling of the problem Hamiltonian. Before embedding the model onto the quantum chip, the coupling strengths ($J_{ij}$) are divided by a rescaling factor $\alpha$:
  $$\alpha := \frac{\beta_{dwave}}{\beta_{unitary}}$$
• This adjustment compensates for hardware noise, thermal effects, and control imperfections, ensuring the generated samples align with the target Boltzmann distribution.

C. Training Algorithm

The RBM training process replaces the classical MCMC sampling step with DQA:

1. Embedding: The RBM weights are mapped to the quantum annealer's qubit connectivity.
2. Sampling: The annealer runs multiple times (5 ns per run) to generate independent spin configurations (samples).
3. Gradient Update: The model parameters are updated using the difference between data expectations and model expectations derived from the quantum samples.

3. Key Contributions

1. First Practical Application of DQA Theory: This is the first experimental demonstration applying the analytic relation between annealing schedules and effective temperature to machine learning training.
2. Systematic Error Mitigation: The paper identifies a systematic temperature misalignment in analog quantum computers and introduces a parameter rescaling method ($\alpha$) to correct it, transforming quantum annealers into principled Boltzmann samplers.
3. Scalability and Connectivity: The approach shifts the complexity of sampling from algorithmic correlation reduction (classical MCMC) to hardware connectivity. This allows the method to naturally extend from sparse RBMs to fully connected Boltzmann Machines, which are intractable for classical sampling.
4. Large-Scale Benchmark: The study trains RBMs on full-resolution MNIST and Fashion-MNIST datasets (784 visible + 1200 hidden units = 1984 qubits) without dimensionality reduction, setting a new benchmark for the scale of quantum machine learning experiments.

4. Results

• Convergence Speed: The DQA-based training converged significantly faster than classical Persistent Contrastive Divergence (PCD).
  • Sampling Time: DQA generated a sample in $\approx 2.56 \times 10^{-4}$ seconds, compared to $\approx 1.63 \times 10^{-2}$ seconds for PCD (with $K=100$ steps). This represents a ~64x speedup in sampling time.
• Validation Error:
  • DQA-trained models achieved lower reconstruction errors (Hamming distance) on both MNIST and Fashion-MNIST compared to PCD.
  • The rescaled DQA (with temperature correction) consistently outperformed uncorrected DQA, proving the necessity of the calibration step.
• Scalability:
  • As the hidden layer size ($N_H$) increased, the performance gap between DQA and PCD widened.
  • Asymptotic Scaling: Classical PCD sampling time scales as $O(N^{2+z})$ (where $z > 0$ is the dynamical critical exponent due to correlation decay), whereas DQA scales as $O(N^2)$ (dominated by static state preparation). This indicates DQA becomes increasingly advantageous for larger models.
• Generative Quality: Visual inspection of generated samples showed that DQA-trained models produced higher-quality images with better structural fidelity.

5. Significance

• Practical Quantum Advantage: The paper demonstrates a clear, practical quantum advantage in a specific machine learning task (sampling for EBM training) by overcoming the historical hurdle of temperature control in quantum annealers.
• New Paradigm for Generative Models: By decoupling the sampling complexity from the algorithmic constraints of MCMC, this method opens the door to training fully connected Boltzmann Machines and other complex generative models (like VAEs) that were previously inaccessible due to sampling intractability.
• Hardware-Agnostic Framework: While demonstrated on D-Wave (superconducting), the theoretical framework and rescaling method are applicable to other analog quantum platforms, including neutral-atom systems.
• Bridge to Gate-Based Models: The authors note that the annealing dynamics can be approximated on gate-based quantum computers via Trotterization, suggesting a pathway for future fault-tolerant implementations.

In conclusion, this work establishes quantum annealers as viable, high-performance Boltzmann samplers for machine learning, provided that systematic hardware temperature deviations are corrected via analytic rescaling.