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

TL;DR

This paper investigates the learnability of shallow quantum circuits using quantum statistical queries, demonstrating their robustness, establishing lower bounds, and showing the limitations of pseudorandom unitaries with constant-depth circuits.

Contribution

It introduces a quantum statistical query framework for learning shallow quantum circuits and proves lower bounds and limitations related to pseudorandom unitaries.

Findings

Quantum statistical queries are robust for learning quantum processes.

Shallow quantum circuits can be learned with linear query complexity overhead.

Constant-depth pseudorandom unitaries cannot be efficiently constructed or distinguished.

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.