Multi-Mode Quantum Annealing for Generative Representation Learning with Boltzmann Priors

Gilhan Kim; Daniel K. Park

quant-phApr 1, 2026

Multi-Mode Quantum Annealing for Generative Representation Learning with Boltzmann Priors

Authors: Gilhan Kim, Daniel K. Park

Organizations: Department of Statistics and Data Science, Yonsei University, Seoul 03722, Republic of Korea · Department of Applied Statistics, Yonsei University, Seoul 03722, Republic of Korea · Department of Quantum Information, Yonsei University, Seoul 03722, Republic of Korea

Abstract

Energy-based models provide a natural bridge between statistical physics and machine learning by representing data through structured energy landscapes. Boltzmann machines are a particularly compelling class of such models for capturing complex interactions among latent variables, but their use in modern generative learning has been limited by the classical intractability of sampling from general (non-restricted) Boltzmann distributions. Here we develop a quantum-annealing-based framework that enables variational autoencoders with general Boltzmann priors. The framework employs three complementary annealing modes tailored to different stages of learning and deployment: diabatic quantum annealing provides unbiased Boltzmann samples for efficient training, slower annealing concentrates samples near low-energy configurations of the learned prior for unconditional generation, and conditional annealing with external fields steers the learned energy landscape toward attribute-specific regions for conditional generation and semantic editing. Using up to 2000 qubits on a D-Wave Advantage2 processor, we demonstrate stable training and high-quality generation on MNIST, Fashion-MNIST, and CelebA, achieving faster convergence and lower reconstruction loss than a Gaussian-prior VAE with the same encoder-decoder architecture. Beyond generation, the learned energy function provides out-of-distribution detection signals that add discriminative power beyond reconstruction loss. We demonstrate that these scores separate in-distribution samples from held-out digit classes in one-class MNIST experiments and improve the detection of market regime shifts in financial data. These results establish quantum annealing as a practical and controllable physical mechanism for energy-based representation learning and generative modeling beyond the reach of tractable classical approaches.

Figures & tables

Figure 1: Schematic illustration of a variational autoencoder with a Boltzmann prior. The encoder maps an input $x$ to a logit vector $l$ . Applying the sigmoid function $\sigma$ componentwise converts these logits into the Bernoulli probabilities that parameterize the approximate posterior $q_{\phi}(z|x)$ over binary latent variables $z\in\{\pm 1\}^{K}$ . A latent sample $z$ drawn from this posterior is passed to the decoder to reconstruct $\tilde{x}$ . The Boltzmann prior $p_{\psi}(z)\propto\exp[-E_{\psi}(z)]$ is trained to match the aggregated posterior $\bar{q}(z)=\mathbb{E}_{x}[q_{\phi}(z|x)]$ . After training, new samples can be generated by drawing $z$ from the learned prior and decoding it through $p_{\theta}(x|z)$ .

Figure 2: Three quantum annealing modes applied to the same learned energy landscape. Blue (DQA): diabatic quantum annealing yields samples that approximately follow a Boltzmann distribution over the landscape and are used for gradient estimation during training. Red (QA): slower quantum annealing localizes samples near low-energy minima for unconditional generation. Green (c-QA): conditional quantum annealing with external bias fields steers sampling toward a specific low-energy region associated with a desired attribute. See Figs. 4 and 5 for generated samples.

Figure 3: Training curves of BM-VAE and Gaussian-prior VAE (G-VAE) on MNIST (left), Fashion-MNIST (center), and CelebA (right). The vertical axis shows the binary cross-entropy reconstruction loss [Eq. ( 7 )]. Solid lines represent the mean over 10 independent runs and shaded regions indicate one standard deviation, where run-to-run variability reflects random parameter initialization and the stochastic nature of quantum annealing samples. For each dataset, the two models share the same encoder–decoder architecture and latent dimensionality and differ only in the choice of latent prior.

Figure 4: Unconditional samples from the learned Boltzmann prior on CelebA ( $128\times 128$ , $K=2000$ latent variables). Samples are generated on the D-Wave Advantage2 processor using QA (Mode 2), which localizes sampling near low-energy minima of the learned energy landscape. No additional denoising or post-processing is applied.

Figure 5: Conditional generation on CelebA using the attribute-average encoder output for Bangs. Row 1: the binarized encoder output $\mathrm{sign}(\bm{l})$ is decoded directly without quantum annealing, producing a single deterministic but visually rigid output. Row 2: c-QA (Mode 3) with the learned couplings $J$ and bias fields $h$ derived from $\bm{l}_{\mathrm{attr}}$ . The pairwise interactions of the Boltzmann prior propagate the attribute bias across latent variables, yielding samples that are both diverse and visually consistent.

Figure 6: Attribute manipulation via c-QA (Mode 3) on CelebA. Left column: original test image. Remaining columns: five independent c-QA samples with Bangs added. For each target attribute, we construct a conditioned logit vector by summing the encoder output of the test image with the attribute-average encoder output. The resulting vector defines the bias fields $h$ used in c-QA together with the learned couplings $J$ . The learned prior produces semantically consistent edits while preserving the identity of the original face, with stochastic diversity across samples.

Figure 7: One-class anomaly detection on MNIST. BM-VAE is trained on a single digit class (normal); all other digit classes are treated as anomalies. Top row: score distributions of normal (blue) and anomalous (red) test samples when digit “1” is the normal class, shown for reconstruction loss (left), Boltzmann energy (center), and posterior entropy (right). Bottom row: per-class AUC when each digit 0–9 serves as the normal class; the red bar highlights digit “1” corresponding to the top row. Dashed lines indicate the mean AUC across all ten classes. Reconstruction loss achieves the highest mean AUC (0.862), while energy (0.720) and entropy (0.599) capture complementary aspects of distributional shift.

	AE	G-VAE	BM-VAE
Recon only	0.793	0.775	0.793
Prior metric	–	0.744	0.757
Rapid-onset episodes
GFC (2008)	0.847	0.846	0.843
Flash Crash	0.906	0.899	0.911
EU Debt	0.951	0.940	0.950

Table 1: Market regime detection performance measured by AUC (

K=1000

). The first two rows report overall AUC using a single score type: reconstruction loss and prior-based metric (energy for BM-VAE, KL divergence for G-VAE), respectively. Episode-level AUC is computed using the best weighted combination of all available scores for each model. Bold indicates best per row.

Figure 8: Score distributions and time series for market regime detection using BM-VAE (

K=1000

). Top row: histograms of normal-period (blue) and anomalous-regime (red) samples for reconstruction loss (left, AUC 0.793), negative expected energy

-A_{E}

(center, AUC 0.757), and posterior entropy (right, AUC 0.699). Bottom rows show each score as a time series from 2005 to 2024: reconstruction loss (second row), energy (third row), and entropy (fourth row). Red shading marks the seven anomalous regime episodes listed in Table 1 and green shading marks the training period (2012–2017). Note that both reconstruction loss and energy are lower during anomalous regimes, reflecting the increased correlation structure of stressed markets (see text).

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