Low-Stabilizer-Complexity Quantum States Are Not Pseudorandom
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We show that quantum states with "low stabilizer complexity" can be efficiently distinguished from Haar-random. Specifically, given an n-qubit pure state |ψ⟩, we give an efficient algorithm that distinguishes whether |ψ⟩ is (i) Haar-random or (ii) a state with stabilizer fidelity at least 1/k (i.e., has fidelity at least 1/k with some stabilizer state), promised that one of these is the case. With black-box access to |ψ⟩, our algorithm uses O( k^12log(1/δ)) copies of |ψ⟩ and O(n k^12log(1/δ)) time to succeed with probability at least 1-δ, and, with access to a state preparation unitary for |ψ⟩ (and its inverse), O( k^3log(1/δ)) queries and O(n k^3log(1/δ)) time suffice. As a corollary, we prove that ω(log(n)) T-gates are necessary for any Clifford+T circuit to prepare computationally pseudorandom quantum states, a first-of-its-kind lower bound.
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