General Phase Matching Condition for Quantum Searching

Abstract

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.