Counting by quantum eigenvalue estimation

Abstract

For every “computation” there corresponds the physical task of manipulating a starting state into an output state with a desired property. As the classical theory of physics has been replaced by quantum physics, it is interesting to consider the capabilities of a computer that can exploit the distinctive quantum features of nature. The extra capabilities seem enormous. For example, with only an expected O( N ) evaluations of a function f:{0,1,…, N−1}→{0,1}, we can find a solution to f( x)=1 provided one exists. Another example is the ability to find efficiently the order of an element g in a group by using a quantum computer to estimate a random eigenvalue of the unitary operator that multiplies by g in the group. By using this eigenvalue estimation algorithm to estimate an eigenvalue of the unitary operator used in quantum searching we can approximately count the number of solutions to f( x)=1. This paper describes this eigenvector approach to quantum counting and related algorithms.