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»promised (i) |ψ»is ϵ-close to a stabilizer state in fidelity or (ii) |ψ»is ϵ-far from all stabilizer states, decide which is the case. We show that for every ϵ>0 and ϵ≤ ϵ^C, there is a poly(1/ϵ)-sample and n· poly(1/ϵ)-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.