Geometric strategy for the optimal quantum search - art. no. 042317

We explore quantum search from the geometric viewpoint of a complex project ive space CP, a space of rays. First. we show that the optimal quantum sear ch can be geometrically identified with the shortest path along the geodesi c joining a target state, an element of the computational basis, and such a n initial state as overlaps equally, up to phases, with all the elements of the computational basis. Second, we calculate the entanglement through the algorithm for any number of qubits n as the minimum Fubini-Study distance to the submanifold formed by separable states in Segre embedding, and find that entanglement is used almost maximally for large n. The computational t ime seems to be optimized by the dynamics as the geodesic, running across e ntangled states away from the submanifold of separable states, rather than the amount of entanglement itself.