A comment on Fisher Information and Quantum Algorithms

We show that Grover’s algorithm defines a geodesic in quantum Hilbert space with the Fubini-Study metric. From statistical point of view Grover’s algorithm is determined by a condition of minima for Fisher’s information function. 1.Recently a lot of attention has been paid to the problem of defining quantum algorithms [1] [2] [3]. Generically a quantum algorithm defines a discrete path in a quantum Hilbert space with the end point of the path corresponding to a quantum state that, after an appropriated measurement, will eventually provide, with high probability, the answer to a given problem. In this note we will work out some geometrical aspects of quantum algorithms. In particular, and taking as example Grover’s algorithm [3], we will show that the path in the quantum Hilbert spaceassociated with the algorithm, is a geodesic in Fubini-Study metric [4] [5] . Geodesics in quantum Hilbert space are intimately connected with Fisher information function [7]. Using Fisher’s function we will define a formal Lagrangian on probability space such that their trajectories coincide with the quantum algorithm path. 2.Grover’s algorithm provides a way to find, by means of a quantum computer [1], one particular item in a set of N items randomly ordered -after approximately π 4 √ N iterations. This algorithm is known to be optimal [8]. In order to define the algorithm let us introduce the quantum state: