On the Exponential Sample Complexity of the Quantum State Sign Estimation Problem
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We demonstrate that the ability to estimate the relative sign of an arbitrary n-qubit quantum state (with real amplitudes), given only k copies of that state, would yield a kn-query algorithm for unstructured search. Thus the quantum sample complexity of sign estimation must be exponential: Ω(2^n/2/n). In particular, we show that an efficient procedure for solving the sign estimation problem would allow for a polynomial time solution to the NP-complete problem 3-SAT.
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