Revisiting Deutsch-Jozsa algorithm

Abstract

The Deutsch-Jozsa algorithm is essentially faster than any possible deterministic classical algorithm for solving a promise problem that is in fact a symmetric partial Boolean function, named as the Deutsch-Jozsa problem. The Deutsch-Jozsa problem can be equivalently described as a partial function DJn0:{0,1}n→{0,1} defined as: DJn0(x)=1 for |x|=n/2, DJn0(x)=0 for |x|=0,n, and it is undefined for the remaining cases, where n is even, and |x| is the Hamming weight of x. The Deutsch-Jozsa algorithm needs only one query to compute DJn0 but the classical deterministic algorithm requires n2+1 queries to compute it in the worse case.We present all symmetric partial Boolean functions with degree 1 and 2; We prove the exact quantum query complexity of all symmetric partial Boolean functions with degree 1 and 2. We prove Deutsch-Jozsa algorithm can compute any symmetric partial Boolean function f with exact quantum 1-query complexity.