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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 thatstate, would yield a $kn$-query algorithm for unstructured search. Thus thequantum sample complexity of sign estimation must be exponential:$\Omega(2^{n/2}/n)$. In particular, we show that an efficient procedure forsolving the sign estimation problem would allow for a polynomial time solutionto the NP-complete problem 3-SAT.

On the Exponential Sample Complexity of the Quantum State Sign Estimation Problem