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We consider the following task: suppose an algorithm is given copies of an unknown n-qubit quantum state | promised (i) | is ₁-close to a stabilizer state in fidelity or (ii) | is ₂-far from all stabilizer states, decide which is the case. We give a poly (1/₁) -sample and n poly (1/₁) -time algorithm for this task for every ₁>0 and ₂ 2^-poly (1/₁). Our proof includes a new definition of Gowers norm for quantum states, an inverse theorem for the Gowers-3 norm of states and new bounds on stabilizer covering for structured subsets of Paulis using results in additive combinatorics.
Arunachalam et al. (Mon,) studied this question.
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