Exact Quantum 1-Query Algorithms and Complexity

Deutsch-Jozsa problem (D-J) has exact quantum 1-query complexity ("exact" means no error), but requires super-exponential queries for the optimal classical deterministic decision trees. D-J problem is equivalent to a symmetric partial Boolean function, and in fact, all symmetric partial Boolean functions having exact quantum 1-query complexity have been found out and these functions can be computed by D-J algorithm. A special case is that all symmetric Boolean functions with exact quantum 1-query complexity follow directly and these functions are also all total Boolean functions with exact quantum 1-query complexity obviously. Then there are pending problems concerning partial Boolean functions having exact quantum 1-query complexity and new results have been found, but some problems are still open. In this paper, we review these results regarding exact quantum 1-query complexity and in particular, we also obtain a new result that a partial Boolean function with exact quantum 1-query complexity is constructed and it cannot be computed by D-J algorithm. Further problems are pointed out for future study.