Generative Quantum Data Embeddings for Supervised Learning
Authors: Jaewoong Heo, 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
Many practically relevant applications of quantum machine learning involve classical data, for which performance depends critically on how inputs are embedded into quantum states. Yet the use of a fixed embedding circuit ansatz remains standard practice. We propose an energy-based generative learning framework that synthesizes gate sequences to optimize embedding structures and refine data-tailored parameters, using a fidelity-based surrogate objective to guide the search toward improved class distinguishability. Empirically, the method improves classification performance across diverse settings, while also revealing datasets where architecture search within the present embedding family yields only limited additional gains. We explain this saturation by deriving bounds on the achievable empirical risk in terms of the Wasserstein distance in the input space, showing that classical data geometry provides an \emph{a priori} diagnostic for regimes in which substantial gains from embedding optimization are unlikely. The results establish a practically useful and theoretically motivated framework for searching effective quantum data embeddings through generative optimization, with the attainable gains diagnosed through the geometry of the underlying classical data.
Autoencoders transformed classical machine learning by solving the curse of dimensionality, enabling principled weight initialization and learning compact, structured representations. In this work, we extend this paradigm to quantum machine learning by introducing a variational autoencoder framework that learns task-specific quantum embeddings of classical data. We demonstrate that high-dimensional datasets, including ImageNet, can be compressed into a 13-qubit quantum representation while remaining reconstructable through a learned decoder. On MNIST (3 vs 5), our approach achieves 98.5% validation accuracy using a circuit-centric quantum classifier, within 1.2 percentage points of a classical neural network baseline (99.7%) and more than 30 percentage points above a naive amplitude-embedding approach. Unlike amplitude embeddings, which require full quantum state tomography for recovery, or angle embeddings, which generally rely on circuit inversion under restrictive assumptions, the proposed framework reconstructs the original data from only a polynomial number of measurements. The framework was further validated on IBM quantum hardware, confirming that the learned embeddings remain stable and reconstructable under real device noise.
Quantum machine learning is often motivated by the idea that quantum systems can expose useful high-dimensional structure that is difficult to access with classical models. We isolate one central component of this claim: the fixed data-encoding map. Amplitude, angle, and basis encoding are evaluated as deterministic feature maps for classical supervised learning under matched output dimensionality and strong classical controls. The benchmark compares these encodings against raw linear models, random Fourier features, polynomial features, PCA, RBF SVMs, and shallow neural networks across diverse classical datasets. Rather than treating performance as a single endpoint, we analyze the geometry of each representation through effective rank, condition number, centered kernel alignment, predictive performance, and practical overhead. The resulting picture is mechanistic: amplitude encoding can remove magnitude information through unit-sphere normalization, angle encoding can become geometrically redundant with raw linear features, and basis encoding can impose a binary Hamming geometry that is poorly aligned with smooth decision structure. These findings do not argue against quantum computation, however, they show that fixed quantum-inspired encoding geometry alone is not a reliable source of machine-learning advantage on classical data.
Generative models have become central across science and industry, from image and text synthesis to the design of molecules and materials. Quantum generative models are considered one of the most promising applications for quantum computers, since a quantum circuit naturally produces samples from the distribution it encodes, and for suitable circuits that distribution is believed to be hard for any classical computer to reproduce. A leading strategy trains these models on a classical computer and reserves the quantum device for generating samples at deployment. This is possible when the training loss can be evaluated on a classical computer. A prime example is the maximum mean discrepancy (MMD2), a moment-matching loss that compares the model and the data through their Pauli-Z correlations. Research so far has asked whether such models can be trained and whether their sampling is hard; whether minimizing such an objective yields a model that generalizes, rather than one that merely reproduces the training statistics, remains poorly understood. We benchmark a broad set of quantum and classical generative models by direct sampling and show that models trained with a moment-matching loss generally show worse generalization than the likelihood-trained models. We show this on two application-inspired datasets: first a cardinality-constrained dataset at up to 30 qubits and second a dataset of genomic single-nucleotide variants, whose valid set is the observed data. These results indicate that a converged moment-matching loss is not a reliable measure of generalization, and that train-classical, deploy-quantum workflows will need approaches that target generalization directly, leaving open whether better training objectives suffice or whether the model architectures themselves must change.