Polynomial-time tolerant testing stabilizer states

We consider the following task: suppose an algorithm is given copies of an unknown $n$-qubit quantum state $|ψ\rangle$ promised $(i)$ $|ψ\rangle$ is $\varepsilon_1$-close to a stabilizer state in fidelity or $(ii)$ $|ψ\rangle$ is $\varepsilon_2$-far from all stabilizer states, decide which is the case. We show that for every $\varepsilon_1>0$ and $\varepsilon_2\leq \varepsilon_1^C$, there is a $\textsf{poly}(1/\varepsilon_1)$-sample and $n\cdot \textsf{poly}(1/\varepsilon_1)$-time algorithm that decides which is the case (where $C>1$ is a universal constant). Our proof includes a new definition of Gowers norm for quantum states, an inverse theorem for the Gowers-$3$ norm of quantum states and new bounds on stabilizer covering for structured subsets of Paulis using results in additive combinatorics.