Quantum statistical query learning

Abstract

We propose a learning model called the quantum statistical learning QSQ model, which extends the SQ learning model introduced by Kearns to the quantum setting. Our model can be also seen as a restriction of the quantum PAC learning model: here, the learner does not have direct access to quantum examples, but can only obtain estimates of measurement statistics on them. Theoretically, this model provides a simple yet expressive setting to explore the power of quantum examples in machine learning. From a practical perspective, since simpler operations are required, learning algorithms in the QSQ model are more feasible for implementation on near-term quantum devices. We prove a number of results about the QSQ learning model. We first show that parity functions, (log n)-juntas and polynomial-sized DNF formulas are efficiently learnable in the QSQ model, in contrast to the classical setting where these problems are provably hard. This implies that many of the advantages of quantum PAC learning can be realized even in the more restricted quantum SQ learning model. It is well-known that weak statistical query dimension, denoted by WSQDIM(C), characterizes the complexity of learning a concept class C in the classical SQ model. We show that log(WSQDIM(C)) is a lower bound on the complexity of QSQ learning, and furthermore it is tight for certain concept classes C. Additionally, we show that this quantity provides strong lower bounds for the small-bias quantum communication model under product distributions. Finally, we introduce the notion of private quantum PAC learning, in which a quantum PAC learner is required to be differentially private. We show that learnability in the QSQ model implies learnability in the quantum private PAC model. Additionally, we show that in the private PAC learning setting, the classical and quantum sample complexities are equal, up to constant factors.

arXiv comment24 Pages. Version 2, minor edits to improve presentation

References in corpus (9)

• Quantum support vector machine for big data classification
• Quantum algorithms for supervised and unsupervised machine learning
• Sampling-based sublinear low-rank matrix arithmetic framework for dequantizing quantum machine learning
• Improved Bounds on Quantum Learning Algorithms
• The Optimal Sample Complexity of PAC Learning
• Learning with Errors is easy with quantum samples
• A Survey of Quantum Learning Theory
• Statistical Queries and Statistical Algorithms: Foundations and Applications
• Entanglement is Necessary for Optimal Quantum Property Testing

Cited by in corpus (13)

• Robust data encodings for quantum classifiers
• Information-theoretic bounds on quantum advantage in machine learning
• Quantum noise protects quantum classifiers against adversaries
• Quantum Differentially Private Sparse Regression Learning
• Flexible learning of quantum states with generative query neural networks
• Exponentially tighter bounds on limitations of quantum error mitigation
• On the Hardness of PAC-learning Stabilizer States with Noise
• Binary Classification with Classical Instances and Quantum Labels
• Private learning implies quantum stability
• On the average-case complexity of learning output distributions of quantum circuits
• Learning unitaries with quantum statistical queries
• Clifford Circuits can be Properly PAC Learned if and only if $\text{RP}=\text{NP}$
• Adversarial Manifold Estimation