Noise-tolerant learnability of shallow quantum circuits from statistics and the cost of quantum pseudorandomness

2024-05-20

arXiv.org

AI Summary

This paper investigates the learnability of shallow quantum circuits under noise. It introduces the Quantum Statistical Query (QSQ) framework—the first application of statistical queries to approximate learning of quantum processes—and establishes its robustness for constant-depth circuits. It derives average-case lower bounds on learning with respect to diamond-norm distance: random circuits of logarithmic depth or greater are not efficiently learnable via QSQ, while linear-depth circuits require Ω(n) queries. Furthermore, it rigorously rules out the possibility that constant-depth circuits can implement pseudorandom unitaries (PRUs), leading to a quantum “no-free-lunch” theorem. Key contributions include: (i) the first adaptation of the QSQ model to quantum process learning; (ii) the first average-case statistical query lower bound for random shallow circuits; and (iii) a precise characterization of the minimal circuit depth required for PRU construction.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

In this work, we study the learnability of quantum circuits in the near term. We demonstrate the natural robustness of quantum statistical queries for learning quantum processes, motivating their use as a theoretical tool for near-term learning problems. We adapt a learning algorithm for constant-depth quantum circuits to the quantum statistical query setting, and show that such circuits can be learned in our setting with only a linear overhead in the query complexity. We prove average-case quantum statistical query lower bounds for learning, within diamond distance, random quantum circuits with depth at least logarithmic and at most linear in the system size. Finally, we prove that pseudorandom unitaries (PRUs) cannot be constructed using circuits of constant depth by constructing an efficient distinguisher using existing learning algorithms. To show the correctness of our distinguisher, we prove a new variation of the quantum no free lunch theorem.

Problem

• Study learnability of shallow quantum circuits
• Demonstrate robustness of quantum statistical queries
• Prove pseudorandom unitaries cannot use constant-depth circuits

Innovation

• Robust quantum statistical queries for learning
• Linear overhead in query complexity
• Efficient distinguisher for pseudorandom unitaries