Phase matching in quantum search algorithm

Abstract

We systematically generalize Grover algorithm in a density matrix formalism by exploiting the underlying two dimensional subspace of the problem. Using this, we derive analytic expressions for the success probability and $l_{1}$ norm of coherence measure $(C_{l_{1}} )$ after arbitrary iterations of the generalized Grover operator with two generic phase angles (α, β). We show for the phase matching condition α = −β = 0.268π with three iterations, success probability ≥ 0.8 can be achieved only with knowledge about the lower bound λ = 0.14, where λ is the ratio of marked to total number of states in the database. This result will improve the quantum search algorithm when applied to databases with unknown number of marked states.