Partial Boolean functions computed by exact quantum 1-query algorithms

Deutsch's algorithm and Deutsch-Jozsa algorithm are exact quantum 1-query algorithms, and in recent years, it has proved that all symmetric partial Boolean functions and total Boolean functions by exact quantum 1-query algorithms can be computed exactly by Deutsch-Jozsa algorithm. Considering the most general case (i.e., all partial Boolean functions), in this paper we obtain four new results: (1) We prove that all partial Boolean functions computed by exact quantum 1-query algorithms can be reduced to a simple form; (2) We discover that all reduced partial Boolean functions computed by exact quantum 1-query algorithms can be represented by degree-1 multilinear polynomials; (3) For small partial Boolean functions up to four bits, we show that there are only 10 new non-trivial reduced partial Boolean functions computed by exact quantum 1-query algorithms; (4) We propose a construction method for finding out all partial Boolean functions computed by a given exact quantum 1-query algorithm. These results break through a basic conclusion that the polynomial degree of all partial Boolean functions computed by exact quantum 1-query algorithms is one or two and pave a way for finding out more problems that have quantum advantages.