Characterization of exact two-query quantum algorithms

Abstract

Quantum query model is a crucial model for quantum computing, where one query to some input variable of a Boolean function f defined on {0,1}n returns the variable value. The exact query complexity, denoted as QE(f), is defined to be the minimum number of queries required to determine the function value. An important problem in this area is to give a succinct characterization of a k-query exact quantum algorithm for an arbitrary k. To date, the cases k=1 and k=n are already solved and the case k=2 remains unknown. Our result is that there are 27 nondegenerate Boolean functions up to isomorphism with QE(f) being two, among which only two functions can be solved by a 2-query classical algorithm. The input bit number n of the above 27 functions ranges from 2 to 6, where the case n≤3 is already proved and the case n=4 is already found by numerically solving semidefinite programming, which is a complete characterization of quantum query algorithm. Assuming the correctness of the numerical result for n=4, we prove that there are four functions in the case n=5, one in the case n=6 and none in the case n≥7. We further show that the 25 functions for which quantum algorithm has advantage over classical algorithm contain essentially only four different structures.