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: $\Omega(2^{n/2}/n)$. 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.

翻译：我们证明,如果能够估计任意以美元计量的量子状态(具有实际振幅)的相对迹象,只要以该状态的复制值为美元,就能产生一个用于无结构搜索的以美元计价的查询算法。因此,标志估计的量子样本复杂性必须是指数化的:$\Omega(2 ⁇ n/2}/n)美元。特别是,我们表明,解决标志估计问题的高效程序将允许对NP-完整的问题3SAT有一个多时制的解决方案。