Machine learning with minimal use of quantum computers: Provable advantages in Learning Under Quantum Privileged Information (LUQPI)

TL;DR

The paper addresses how to achieve provable quantum advantages with minimal quantum involvement by introducing LUQPI, where a quantum feature extractor processes each data point during training only. It formalizes LUQPI within the LUPI framework, defines online and offline (fully offline) variants, and proves exponential quantum-classical separations for carefully constructed concept classes under cryptographic assumptions, including DDH-based hardness. A concrete ElGamal-encrypted-key construction (EEK) demonstrates the theoretical separation, and a practical Rydberg-chain phase-identification study shows consistent gains for LUQPI-style models when privileged quantum features are used during training. The work highlights that quantum resources can meaningfully augment classical learning even when deployment remains purely classical, suggesting practical avenues for near-term quantum-assisted learning in data-scarce or phase-transition–driven domains.

Abstract

Quantum machine learning (QML) is often listed as a promising candidate for useful applications of quantum computers, in part due to numerous proofs of possible quantum advantages. A central question is how small a role quantum computers can play while still enabling provable learning advantages over classical methods. We study an especially restricted setting in which a quantum computer is used only as a feature extractor: it acts independently on individual data points, without access to labels or global dataset information, is available only to augment the training set, and is not available at deployment. Training and deployment are therefore carried out by fully classical learners on a dataset augmented with quantum-generated features. We formalize this model by adapting the classical framework of Learning Under Privileged Information (LUPI) to the quantum case, which we call Learning Under Quantum Privileged Information (LUQPI). Within this framework, we show that even such minimally involved quantum feature extraction, available only during training, can yield exponential quantum-classical separations for suitable concept classes and data distributions under reasonable computational assumptions. We further situate LUQPI within a taxonomy of related quantum and classical learning settings and show how standard classical machinery, most notably the SVM+ algorithm, can exploit quantum-augmented data. Finally, we present numerical experiments in a physically motivated many-body setting, where privileged quantum features are expectation values of observables on ground states, and observe consistent performance gains for LUQPI-style models over strong classical baselines.

Figures (3)

• Figure 1: Comparison of online and offline feature extraction settings. The training phase is identical for both settings, using quantum feature extraction. In deployment, an online setting requires quantum resources while an offline setting uses purely classical computation. Raw input $x'$ is passed directly to the trained model along with extracted features.
• Figure 2: Model performance comparison across sampling strategies. Left panels show test accuracy for SVM (blue), SVM+ (purple), and Transformer (orange) as functions of training set size. Right panels show SVM+ performance gain over SVM with 95% confidence intervals (green shading indicates improvement).
• Figure 3: Misclassification patterns in hard boundary sampling with 30 training samples (black circles). Ground truth (top left) compared to Standard SVM (top right), SVM+ (bottom left), and Transformer (bottom right). Red X marks indicate misclassified test points.

Theorems & Definitions (26)

• Definition 2.1: Efficient (classical and quantum) PAC learnability
• Definition 2.2: Extended example oracle
• Definition 2.3: Derived/effective hypothesis class induced by $\mathcal{E}$
• Definition 2.4: Quantum online advantageous feature extraction
• Definition 2.5: Quantum-offline advantageous feature extraction
• Theorem 5.1
• Definition 5.1: Deterministic Group Generation
• Definition 5.2: ElGamal Encrypted Key (EEK) Concept Class
• Definition 5.3: Concept-friendly embedding
• Theorem 5.2