量子搜索的一般相位匹配条件
General Phase Matching Condition for Quantum Searching
摘要：
给出了任意幺正变换和任意相位旋转的量子搜索算法的一般相位匹配条件。我们通过一个明确的表达式表明，相位匹配条件既依赖于幺正变换U，也依赖于初始状态。假设初始振幅分布为任意叠加sinθ_0| 1> + cosθ_0 e^{iδ}| 2>具有|1> = {1 / sinβ} \sum_k| τ_k> <τ_k |U| 0>且|2> = {1 / cosβ} \sum_{i \ne τ}| i><i |U| 0>，其中|τ_k>是一个标记态，\sin β= \sqrt{\sum_k|U_{τ_k 0}|^2}由U之间的么正变换U的矩阵元决定|τ_k>和|0>状态，则一般的相位匹配条件为tan{θ/ 2} [cos 2β+ tanθ_0 cosδsin 2β]= tan{θ/ 2} [1-tanθ_0 sinδsin 2βtan{θ/ 2}]，其中θ和θ是相位旋转角，|0>和|τ_k>分别。这推广了以前的结论，其中相位匹配条件依赖于$U$和初始状态已被掩盖。我们表明，以前在文献中讨论的几个相位条件是这个一般的情况下，这澄清了问题的条件应被视为准确的特殊情况。
英文摘要：
We present a general phase matching condition for the quantum search algorithm with arbitrary unitary transformation and arbitrary phase rotations. We show by an explicit expression that the phase matching condition depends both on the unitary transformation U and the initial state. Assuming that the initial amplitude distribution is an arbitrary superposition sinθ_0 |1> + cosθ_0 e^{iδ} |2> with |1> = {1 / sinβ} \sum_k |τ_k> <τ_k|U|0> and |2> = {1 / cosβ} \sum_{i \ne τ}|i> <i|U|0>, where |τ_k> is a marked state and \sinβ= \sqrt{\sum_k|U_{τ_k 0}|^2} is determined by the matrix elements of unitary transformation U between |τ_k> and the |0> state, then the general phase matching condition is tan{θ/ 2} [cos 2β+ tanθ_0 cosδsin 2β]= tan{ϕ/ 2} [1-tanθ_0 sinδsin 2βtan{θ/ 2}], where θand ϕare the phase rotation angles for |0> and |τ_k>, respectively. This generalizes previous conclusions in which the dependence of phase matching condition on $U$ and the initial state has been disguised. We show that several phase conditions previously discussed in the literature are special cases of this general one, which clarifies the question of which condition should be regarded as exact.
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