Polynomial-time tolerant testing stabilizer states

We consider the following task: suppose an algorithm is given copies of an unknown n-qubit quantum state |psirangle promised (i) |psirangle is varepsilon1-close to a stabilizer state in fidelity or (ii) |psirangle is varepsilon2-far from all stabilizer states, decide which is the case. We show that for every varepsilon1>0 and varepsilon2≤ varepsilon1C, there is a textsfpoly(1/varepsilon1)-sample and n· textsfpoly(1/varepsilon1)-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.