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Analysis of Generalized Grover's Quantum Search Algorithms Using Recursion Equations

The recursion equation analysis of Grover's quantum search algorithm presented by Biham et al. [PRA 60, 2742 (1999)] is generalized. It is applied to the large class of Grover's type algorithms in which the Hadamard transform is replaced by any other unitary transformation and the phase inversion is replaced by a rotation by an arbitrary angle. The time evolution of the amplitudes of the marked and unmarked states, for any initial complex amplitude distribution is expressed using first order linear difference equations. These equations are solved exactly. The solution provides the number of iterations T after which the probability of finding a marked state upon measurement is the highest, as well as the value of this probability, P_max. Both T and P_max are found to depend on the averages and variances of the initial amplitude distributions of the marked and unmarked states, but not on higher moments.