Single-copy stabilizer testing

TL;DR

This paper presents an efficient algorithm for testing whether an unknown n-qubit state is a stabilizer state using O(n) copies, and proves a lower bound of Ω(√n) copies for any such testing method.

Contribution

It introduces a new algorithm for stabilizer state testing with optimal sample complexity and establishes a matching lower bound, advancing understanding of quantum state property testing.

Findings

Algorithm uses O(n) copies for stabilizer testing.

Any algorithm requires at least Ω(√n) copies.

Stabilizer states are most likely to show linear dependencies in random stabilizer basis measurements.

Abstract

We consider the problem of testing whether an unknown $n$-qubit quantum state $|\psi\rangle$ is a stabilizer state, with only single-copy access. We give an algorithm solving this problem using $O(n)$ copies, and conversely prove that $\Omega(\sqrt{n})$ copies are required for any algorithm. The main observation behind our algorithm is that when repeatedly measuring in a randomly chosen stabilizer basis, stabilizer states are the most likely among the set of all pure states to exhibit linear dependencies in measurement outcomes. Our algorithm is designed to probe deviations from this extremal behavior. For the lower bound, we first reduce stabilizer testing to the task of distinguishing random stabilizer states from the maximally mixed state. We then argue that, without loss of generality, it is sufficient to consider measurement strategies that a) lie in the commutant of the tensor action of the Clifford group and b) satisfy a Positive Partial Transpose (PPT) condition. By leveraging these constraints, together with novel results on the partial transposes of the generators of the Clifford commutant, we derive the lower bound on the sample complexity.