Phase matching condition for quantum search with a generalized initial state

We present a general phase matching condition for the quantum search algorithm with arbitrary unitary transformation and arbitrary phase rotations in the case of a general initial state. We show by an explicit expression that the phase matching condition depends both on the unitary transformation U and the initial state. 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.  2002 Elsevier Science B.V. All rights reserved. PACS: 03.67.Lx; 03.67.Hk; 89.70.+c