Analysis of Grover’s quantum search algorithm as a dynamical system

Grover’s quantum search algorithm is analyzed for the case in which the initial state is an arbitrary pure quantum state [Formula Presented] of n qubits. It is shown that the optimal time to perform the measurement is independent of [Formula Presented] namely, it is identical to the optimal time in the original algorithm in which [Formula Presented] with the same number of marked states, r. The probability of success [Formula Presented] is obtained in terms of the amplitudes of the state [Formula Presented] and is shown to be independent of r. A class of states, which includes fixed points and cycles of the Grover iteration operator, is identified. The relevance of these results in the context of using the success probability as an entanglement measure is discussed. In particular, the Groverian entanglement measure, previously limited to a single marked state, is generalized to the case of several marked states.