Sample Complexity of Learning Parametric Quantum Circuits

TL;DR

This paper establishes that parametric quantum circuits with bounded gates are PAC learnable on quantum computers, providing explicit bounds on sample complexity and constructing representative circuit families for quantum machine learning.

Contribution

It proves PAC learnability of physical quantum circuits with explicit sample complexity bounds and constructs a universal variational quantum circuit family.

Findings

Sample complexity is bounded by O(n^{c+1}) for circuits with at most n^c gates.

Constructs a family of variational quantum circuits with O(n^{c+1}) gates.

Provides theoretical guidance for quantum machine learning applications.

Abstract

Quantum computers hold unprecedented potentials for machine learning applications. Here, we prove that physical quantum circuits are PAC (probably approximately correct) learnable on a quantum computer via empirical risk minimization: to learn a parametric quantum circuit with at most $n^c$ gates and each gate acting on a constant number of qubits, the sample complexity is bounded by $\tilde{O}(n^{c+1})$. In particular, we explicitly construct a family of variational quantum circuits with $O(n^{c+1})$ elementary gates arranged in a fixed pattern, which can represent all physical quantum circuits consisting of at most $n^c$ elementary gates. Our results provide a valuable guide for quantum machine learning in both theory and practice.