General Phase Matching Condition for Quantum Searching

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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 sinθ₀ |1> + cosθ₀ eⁱδ |2> with |1> = 1 / sinβ ∑ₖ |τₖ> <τₖ|U|0> and |2> = 1 / cosβ ∑ᵢ ≠ τ|i> <i|U|0>, where |τₖ> is a marked state and β= √∑ₖ|Uτₖ ₀|² is determined by the matrix elements of unitary transformation U between |τₖ> and the |0> state, then the general phase matching condition is tanθ/ 2 [cos 2β+ tanθ₀ cosδsin 2β]= tanϕ/ 2 [1-tanθ₀ sinδsin 2βtanθ/ 2], where θand ϕare the phase rotation angles for |0> and |τₖ>, 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.