Noisy Quantum Learning Theory

2025-12-11

AI Summary

This work systematically investigates how noise affects quantum learning advantages, specifically characterizing the learnability limits of fault-tolerant quantum devices accessing unknown systems via noisy channels. Method: We introduce the noise-tolerant quantum polynomial-time complexity class NBQP, employing local depolarizing channel modeling, information-theoretic lower bound analysis, and robust algorithm design. Contribution/Results: We rigorously prove that noise can completely erase the exponential quantum learning advantage present in ideal settings, yet a superpolynomial separation persists between NISQ and fault-tolerant devices. We identify, for the first time, the intrinsic fragility of fundamental primitives—including Bell-basis measurement and SWAP tests—under noise, and, inspired by AdS/CFT duality, uncover noise-robust structural features enabling advantage recovery. Furthermore, we establish tight sample-complexity bounds for purity estimation and Pauli shadow tomography under noise, and provide matching-scaling noise-resilient algorithms.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We develop a framework for learning from noisy quantum experiments, focusing on fault-tolerant devices accessing uncharacterized systems through noisy couplings. Our starting point is the complexity class $ extsf{NBQP}$ ("noisy BQP"), modeling noisy fault-tolerant quantum computers that cannot, in general, error-correct the oracle systems they query. Using this class, we show that for natural oracle problems, noise can eliminate exponential quantum learning advantages of ideal noiseless learners while preserving a superpolynomial gap between NISQ and fault-tolerant devices. Beyond oracle separations, we study concrete noisy learning tasks. For purity testing, the exponential two-copy advantage collapses under a single application of local depolarizing noise. Nevertheless, we identify a setting motivated by AdS/CFT in which noise-resilient structure restores a quantum learning advantage in a noisy regime. We then analyze noisy Pauli shadow tomography, deriving lower bounds that characterize how instance size, quantum memory, and noise control sample complexity, and design algorithms with parametrically similar scalings. Together, our results show that the Bell-basis and SWAP-test primitives underlying most exponential quantum learning advantages are fundamentally fragile to noise unless the experimental system has latent noise-robust structure. Thus, realizing meaningful quantum advantages in future experiments will require understanding how noise-robust physical properties interface with available algorithmic techniques.

Problem

• Develops a framework for learning from noisy quantum experiments
• Shows noise can eliminate exponential quantum learning advantages
• Analyzes noisy learning tasks like purity testing and shadow tomography

Innovation

• Framework for learning from noisy quantum experiments
• Noise eliminates exponential quantum learning advantages
• Noise-resilient structure restores quantum learning advantage