Analysis of generalized Grover quantum search algorithms using recursion equations - art. no. 012310

The recursion equation analysis of Grover's quantum search algorithm presen ted by Biham et al. [Phys. Rev. A 60, 2742 (1999)] is generalized. It is ap plied to the large class of Grover-type algorithms in which the Hadamard tr ansform is replaced by any other unitary transformation and the phase inver sion 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 eq uations. These equations are solved exactly. The solution provides the numb er of iterations T after which the probability of finding a marked slate up on 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.