Quantum-Assisted Learning of Hardware-Embedded Probabilistic Graphical Models

Marcello Benedetti; John Realpe-Gómez; Rupak Biswas; Alejandro Perdomo-Ortiz

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Quantum-Assisted Learning of Hardware-Embedded Probabilistic Graphical Models

Marcello Benedetti, John Realpe-Gómez, Rupak Biswas, Alejandro Perdomo-Ortiz

arXiv:1609.02542v4Sep 8, 2016quant-phcs.LG

TL;DR

Quantum annealers face sparse connectivity and uncertain, noisy effective parameters when used to sample distributions for generative learning. The paper combines redundant hardware embedding with gray-box, data-driven training, demonstrating robust generative modeling while avoiding per-iteration temperature estimation.

Problem
Quantum annealers’ sparse connectivity, limited controls, noise, and effective-parameter uncertainty constrain their use for generative unsupervised learning.
Method
The approach redundantly embeds logical variables across physical-qubit subgraphs and learns hardware parameters from data using a gray-box model.
Results
The trained hardware-embedded model improves generative performance with training and converges to the ideal value on tested configurations.
Takeaways & Limitations
The results provide experimental evidence that quantum annealers can be trained robustly and used as samplers for generative unsupervised-learning models.
Takeaways & Limitations
The demonstrated model is visible-only, while hidden units and continuous variables remain natural extensions for constraints and real data.

Abstract

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Mainstream machine-learning techniques such as deep learning and probabilistic programming rely heavily on sampling from generally intractable probability distributions. There is increasing interest in the potential advantages of using quantum computing technologies as sampling engines to speed up these tasks or to make them more effective. However, some pressing challenges in state-of-the-art quantum annealers have to be overcome before we can assess their actual performance. The sparse connectivity, resulting from the local interaction between quantum bits in physical hardware implementations, is considered the most severe limitation to the quality of constructing powerful generative unsupervised machine-learning models. Here we use embedding techniques to add redundancy to data sets, allowing us to increase the modeling capacity of quantum annealers. We illustrate our findings by training hardware-embedded graphical models on a binarized data set of handwritten digits and two synthetic data sets in experiments with up to 940 quantum bits. Our model can be trained in quantum hardware without full knowledge of the effective parameters specifying the corresponding quantum Gibbs-like distribution; therefore, this approach avoids the need to infer the effective temperature at each iteration, speeding up learning; it also mitigates the effect of noise in the control parameters, making it robust to deviations from the reference Gibbs distribution. Our approach demonstrates the feasibility of using quantum annealers for implementing generative models, and it provides a suitable framework for benchmarking these quantum technologies on machine-learning-related tasks.

I. INTRODUCTION

Quantum annealers may serve as sampling engines for unsupervised learning, but sparse connectivity, parameter uncertainty, and noise complicate their use. The paper introduces hardware-embedded graphical models and gray-box training to address these challenges and demonstrates generative modeling on several data sets.

Sampling from high-dimensional distributions underlies important machine-learning methods, while unsupervised learning remains comparatively difficult.
Quantum annealers can sample Gibbs-like distributions, but effective temperature and transverse-field parameters depend on programmed controls and may be unknown.
The proposed approach avoids estimating temperature at each learning iteration and uses hardware-embedded pairwise interactions learned directly from data.
The model is a visible-only quantum Boltzmann machine and models binarized handwritten digits, bars-and-stripes, and Sherrington-Kirkpatrick configurations.
Training uses a gray-box framework intended to tolerate deviations from Gibbs sampling, control-parameter noise, and sampling biases without requiring full distribution knowledge.
The paper presents experimental evidence that quantum annealers can be trained robustly for generative unsupervised learning, rather than establishing quantum speedup.

B. Enhancing modeling capacity

Hardware embedding converts a logical graphical model into a physical model whose parameters must be learned for the machine-learning task rather than merely preserving an optimization ground state. This gray-box formulation lets data guide both logical and auxiliary hardware parameters.

Hardware-embedded models represent arbitrary-connectivity logical graphs using subgraphs of the physical interaction graph.
Unlike optimization, generative learning requires matching probabilities across configurations, so standard ground-state-preserving embeddings can implement the wrong distribution.
The parameter-setting problem is complicated by limited precision, noise, and instance-dependent effective temperature during learning.

C. Parameter-setting problem

The method addresses parameter setting by redundantly encoding logical variables across physical-qubit subgraphs and learning the resulting hardware model directly from an extended data set. Majority decoding maps generated physical samples back to logical variables.

In machine learning, hardware parameters are themselves learned because all configuration probabilities matter, unlike optimization where the ground state is the target.
Embedding parameters can be data-guided rather than fixed by prior analytical rules, allowing the quantum annealer to be treated as a trainable neural network.
The approach focuses on effective all-to-all logical connectivity while also representing simpler pairwise topologies with fewer qubits.
Embedding several qubits per logical variable transforms the data into a higher-resolution redundant representation that increases modeling capacity despite sparse hardware connectivity.
Each logical variable is replicated across the qubits of its corresponding hardware subgraph to form the extended training data set.
Generated physical samples are decoded by assigning each logical variable the majority state among its representative physical qubits.

III. LEARNING ALGORITHM

The learning algorithm trains a hardware-embedded quantum maximum-entropy model using data and model moments, avoiding per-iteration temperature estimation. Its adaptive couplings can tolerate certain deviations from the assumed Gibbs-like sampling behavior, although regularization remains unresolved and the quantum regime is not numerically validated.

The model is trained by minimizing quantum relative entropy, using differences between data-dependent and model-dependent first- and second-order moments.The data ensemble is the positive phase, while the model ensemble is the negative phase.
The update rule adjusts inter- and intra-subgraph couplings and local fields with a learning rate η based on estimated moment differences.The learning procedure is derived for the hardware-embedded graphical model and uses iterative parameter updates.
The approach avoids estimating the effective temperature at every iteration and learns intra-subgraph couplings from model–data correlation contrasts rather than hard-coded values.This directly addresses parameter-setting limitations in hardware-embedded models.
Quantum annealer samples need not exactly follow a Gibbs-like distribution if estimated gradients retain a positive projection toward the true gradient.When the transverse field is negligible, the samples are expected to approximately follow a classical Boltzmann distribution.
Intra-subgraph couplings can grow indefinitely in principle, making regularization or a generalization-based stopping criterion necessary; regularization remains an open research question.The issue is described as overfitting caused by gradient descent pushing parameters toward distributions concentrated on the data.
The embedded fully connected model uses O(N^2) qubits and O(N^2) parameters, with experiments reporting an additional-parameter factor of about 3.The qubit overhead arises from representing dense logical graphs on sparse hardware connectivity.

A. Device and embeddings

Experiments use a partially functional DW2X quantum annealer with Chimera connectivity and heuristic minor embeddings. The embedding design is evaluated by both physical-qubit usage and the largest subgraph size.

The DW2X device has 1152 qubits, of which 1097 are functional and available, limiting the size of embeddable fully connected models.With all qubits available, an efficient schema could embed up to 48 logical variables; the reduced availability lowers that maximum.
Embeddings are generated with D-Wave’s find embedding heuristic over 500 requested trials, selecting solutions by qubit count and maximum subgraph size.The procedure prefers embeddings that use fewer physical qubits and have smaller logical-unit subgraphs.
A 46-variable fully connected graph was embedded using 917 qubits with a maximum subgraph size of 34 in one reported embedding.The passage contrasts this with another embedding choice but does not provide its complete characteristics.
The DW2X parameter range is J(kl)ij ∈ [−2, +2], and parameters were initialized in [−10^-6, +10^-6] to break symmetry.The initialization uses small values near zero before learning begins.
Table I reports minimum and maximum subgraph sizes, chip usage, and logical-versus-physical parameter counts for the embeddings.All embeddings were generated with the find embedding heuristic.

B. Data sets and preprocessing

The experiments combine real and synthetic binary datasets adapted to hardware limits. OptDigits is cropped, reduced to four digit classes, and binarized, while BAS and random Sherrington–Kirkpatrick instances provide synthetic benchmarks.

The study tests OptDigits, BAS, and synthetic datasets generated from random Ising instances.These datasets cover a real handwritten-digit task and two synthetic settings.
OptDigits images are cropped from 8 × 8 to 7 × 6 pixels, retaining readability while reducing the embedding burden.The removed regions are the leftmost and rightmost columns and the bottom row.
OptDigits is restricted to digits one through four, reducing the one-hot class encoding to four variables.The original fully connected representation would require 74 variables and exceed the available DW2X embedding capacity.
The BAS dataset uses randomly chosen black or white row and column patterns and is sized to match an embedding of a 42-variable fully connected graph.A 7 × 6 BAS dataset contains 192 pictures of 42 binary variables each.
The four-bit OptDigits grayscale is thresholded at its midpoint and binarized to {−1, +1}.This preprocessing follows the spatial cropping step.
The synthetic Ising benchmark contains 10 random Sherrington–Kirkpatrick instances with N = 15 logical variables, β = 1, ζ = 2, and zero local fields.The instances were selected to produce nontrivial overlap distributions and permit exhaustive computations.
Table II summarizes dataset variable counts, training-point counts, and applicable test-point counts, marking synthetic datasets with an asterisk.The table organizes the basic sizes of the datasets used in the experiments.

C. Choice of hyperparameters

The experiments use separate device and gradient hyperparameters: device settings prioritize fast sampling, while gradient settings use regularization and momentum to improve learning and generalization.

Device hyperparameters: Device hyperparameters are set to their minimum values to obtain samples as quickly as possible.These settings affect the time required to obtain samples.
Gradient hyperparameters: L2 regularization is applied to pairwise interactions, and momentum is applied to all parameters.Both techniques introduce gradient hyperparameters into the learning equations.
Gradient hyperparameters: A small grid search selects gradient-hyperparameter values that produce visually appealing samples.The experiments use the selected hyperparameter settings except where gradient hyperparameters are tuned separately.

V. RESULTS

The hardware-embedded model learns to reconstruct corrupted OptDigits and BAS pictures from uncorrupted pixels, with BAS reconstruction reaching almost perfect accuracy on held-out test data.

Reconstruction procedure: The reconstruction task clamps each correct pixel across its corresponding qubit subgraph before sampling the missing values.Strong local fields enforce the observed pixel values on the hardware embedding.
OptDigits: OptDigits were trained for 6000 iterations using minibatches of 200 data points and 200 DW2X samples per iteration.Training stopped when an intra-subgraph coupling exceeded the device’s dynamic range.
OptDigits: 50% salt-and-pepper corruption still produced qualitatively good OptDigits reconstructions from early training stages.Because multiple completions can be correct, the study visually inspected reconstructions rather than computing an error measure.
Bars and stripes: 0.65% mistaken pixels was achieved after 3850 iterations when reconstructing BAS test images with a blacked-out 5 × 4 block.The error declined from 50% after one iteration to 18.6% after 100, 2.95% after 1000, and 0.65% at the end of training.
Bars and stripes: The BAS test-set reconstructions provide evidence that the pixel distribution was modeled rather than simply memorized.The test pictures were never used during training.

B. Generation and classification of pictures

The paper evaluates hardware-embedded models through class-conditional generation, classification, picture reconstruction, and synthetic-distribution learning, while also comparing physical and logical training strategies.

B. Generation and classification of pictures: The 46-variable OptDigits model adds four one-hot class variables to the pixel representation for generation and classification.The model includes the four selected digit classes alongside the image variables.
B. Generation and classification of pictures: 90% test-picture classification accuracy was obtained by sampling the four class variables after 6000 learning iterations.The procedure uses 100 samples per picture and assigns the most frequent class; random guessing gives 25% accuracy.
B. Generation and classification of pictures: Machine-generated digits were remarkably similar to human test-set digits while also capturing ambiguity from low-resolution calligraphy styles.The visual comparison fixes one class per column and separates human-generated from machine-generated rows.
B. Generation and classification of pictures: BAS reconstructions improved from 50% to 0.65% mistaken pixels across training, indicating strong recovery of held-out picture regions.The evaluation masks a 5 × 4 block while retaining enough unmasked information for reconstruction.
C. Learning of an Ising model: The synthetic comparison samples 150 points from the learned model and evaluates their average log-likelihood under the fully known generating distribution.This proxy is related to generalization but requires knowledge of the true distribution and is therefore infeasible for real data sets.
C. Learning of an Ising model: SA learning on the physical graph was slower than exact-gradient learning on the logical graph, although both approached the data-set optimum.The comparison used 500 iterations on synthetic distributions, with independent linear schedules for samples.

VI. CONCLUSIONS

The paper addresses key experimental barriers to quantum-annealer generative modeling by combining hardware embedding with gray-box, data-driven learning. Experiments support robust training and generative modeling, while quantum speedup, broader distributions, and richer variables remain open directions.

Contributions: The approach uses hardware embedding and data-driven parameter setting to improve the robustness and complexity of models realizable on quantum annealers.The learning rules are embedding-agnostic and can extend to alternative architectures such as LHZ.
Contributions: The gray-box framework avoids estimating effective temperature during learning and may mitigate noise in device control parameters.It requires only partial information about the annealer’s actual sampling distribution.
Results: The embedded visible-only quantum Boltzmann machine represents all pairwise interactions among logical variables and was validated on image and synthetic data.The experiments included reconstruction and generation of pictures and quantitative proxies from randomly generated Boltzmann distributions.
Results: Under matched hyperparameters, the quantum-assisted approach learns faster initially, although the source does not establish quantum speedup over classical methods.The faster initial learning may reflect automatic adjustment of the learning rate through an instance-dependent effective temperature.
Open questions: Important scope boundaries remain: the study does not settle when quantum annealers outperform classical MCMC, and extending the model to hidden or continuous variables remains future work.The paper also leaves open numerical simulation of the annealers’ open-quantum-system dynamics.

Appendix A: Example

The appendix illustrates an embedding of a fully connected logical graph into the hardware graph. Forty-six logical variables are represented using 940 physical qubits, occupying most of the device.

Embedding: 46 logical variables are embedded into 940 physical variables of DW2X’s chimera graph.Qubits representing the same logical variable share an identifier and are connected by same-colored edges.
Embedding: The embedding uses 86% of DW2X’s available qubits.