Characterizations of symmetrically partial Boolean functions with exact quantum query complexity

An {\it exact} quantum algorithm requires its output to be correct certainly, and this kind of algorithms has been studied mostly in the query model since the discoveries of Deutsch-Jozsa algorithm and the solution of Simon's problem. In this article, we prove the optimal exact quantum query complexity for a generalized Deutsch-Jozsa problem and characterize the symmetrically partial Boolean functions in terms of exact quantum query complexity. More specifically, (1) we give an optimal exact quantum query algorithm with complexity $k+1$ for computing the symmetrically partial Boolean function $DJ_n^k$ defined as: $DJ_n^k(x)=1$ for $|x|=n/2$, $DJ_n^k(x)=0$ for $|x|$ in the set $\{0, 1,\ldots, k, n-k, n-k+1,\ldots,n\}$, and it is undefined for the rest cases, where $n$ is even, $|x|$ is the Hamming weight of $x$. The case of $k=0$ is the well-known Deutsch-Jozsa problem. (2) We prove that any symmetrically partial Boolean function $f$ has exact quantum 1-query complexity if and only if $f$ can be computed by the Deutsch-Jozsa algorithm. (3) We also discover the optimal exact quantum 2-query complexity for solving a variant of the Deutsch-Jozsa problem, i.e. distinguishing between inputs of Hamming weight $\{ \lfloor n/2\rfloor, \lceil n/2\rceil \}$ and Hamming weight in the set $\{ 0, n\}$ for all odd $n$. We further discuss the optimal exact quantum algorithms for distinguishing between inputs of two different Hamming weights $k$ and $l$. (4) We provide an algorithm to determine the degree of any symmetrically partial Boolean function.