A note on polynomial-time tolerant testing stabilizer states

TL;DR

This paper improves the understanding of quantum stabilizer states by establishing a stronger inverse theorem for the Gowers-3 norm, leading to an efficient tolerant testing algorithm that distinguishes stabilizer states from non-stabilizer states with high confidence.

Contribution

It provides a new inverse theorem for the Gowers-3 norm of quantum states and develops a polynomial-time tolerant testing algorithm for stabilizer states.

Findings

Enhanced inverse theorem for Gowers-3 norm of quantum states

Polynomial-time tolerant testing algorithm for stabilizer states

Ability to distinguish stabilizer states with high fidelity accuracy

Abstract

We show an improved inverse theorem for the Gowers-$3$ norm of $n$-qubit quantum states $|\psi\rangle$ which states that: for every $\gamma\geq 0$, if the $\textsf{Gowers}(|\psi \rangle,3)^8 \geq \gamma$ then the stabilizer fidelity of $|\psi\rangle$ is at least $\gamma^C$ for some constant $C>1$. This implies a constant-sample polynomial-time tolerant testing algorithm for stabilizer states which accepts if an unknown state is $\varepsilon_1$-close to a stabilizer state in fidelity and rejects when $|\psi\rangle$ is $\varepsilon_2 \leq \varepsilon_1^C$-far from all stabilizer states, promised one of them is the case.