Algebraic analysis of quantum search with pure and mixed states

Abstract

An algebraic analysis of Grover's quantum search algorithm is presented for the case in which the initial state is an arbitrary pure quantum state $\ensuremath{\mid}\ensuremath{\psi}⟩$ of $n$ qubits. This approach reveals the geometrical structure of the quantum search process, which turns out to be confined to a four-dimensional subspace of the Hilbert space. It unifies and generalizes earlier results on the time evolution of the amplitudes during the search, the optimal number of iterations, and the success probability. Furthermore, it enables a direct generalization to the case in which the initial state is a mixed state, providing an exact formula for the success probability.