Polynomial-Time Tolerant Testing Stabilizer States
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Abstract
We consider the following task: suppose an algorithm is given copies of an unknown n-qubit quantum state |ψ⟩ promised (i) |ψ⟩ is ε1-close to a stabilizer state in fidelity or (ii) |ψ⟩ is ε2-far from all stabilizer states, decide which is the case. We show that for every ε1>0 and ε2≤ ε1C, there is a poly(1/ε1)-sample and n· poly(1/ε1)-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.