A note on polynomial-time tolerant testing stabilizer states

Abstract

We show an improved inverse theorem for the Gowers-$3$ norm of $n$-qubit quantum states $|\psi ⟩$ which states that: for every $\gamma \ge 0$, if the $\text{Gowers}(|\psi ⟩,3{)}^8\ge \gamma$ then the stabilizer fidelity of $|\psi ⟩$ 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 ${\epsilon }_1$-close to a stabilizer state in fidelity and rejects when $|\psi ⟩$ is ${\epsilon }_2\le {\epsilon }_1^C$-far from all stabilizer states, promised one of them is the case.

6 pages