Noise-tolerant learnability of shallow quantum circuits from statistics and the cost of quantum pseudorandomness
Chirag Wadhwa, Mina Doosti
TL;DR
This work investigates near-term learnability of quantum circuits through quantum statistical queries (QSQ) and quantum statistical process queries (QPSQ). It introduces new multi-copy oracles and an observable-learning oracle, demonstrates noise-tolerant learning for constant-depth circuits with a linear overhead, and provides average-case lower bounds for learning shallow random circuits. A key result shows that constant-depth circuits cannot implement pseudorandom unitaries, establishing a depth cost for PRUs via a learning-and-verification distinguisher. Collectively, the paper advances robust, noise-aware theoretical foundations for learning quantum processes while clarifying depth requirements for secure pseudorandomness in quantum circuits.
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.