Partial Boolean functions with exact quantum 1-query complexity

We provide two sufficient and necessary conditions to characterize any $n$-bit partial Boolean function with exact quantum 1-query complexity. Using the first characterization, we present all $n$-bit partial Boolean functions that depend on $n$ bits and have exact quantum 1-query complexity. Due to the second characterization, we construct a function $F$ that maps any $n$-bit partial Boolean function to some integer, and if an $n$-bit partial Boolean function $f$ depends on $k$ bits and has exact quantum 1-query complexity, then $F(f)$ is non-positive. In addition, we show that the number of all $n$-bit partial Boolean functions that depend on $k$ bits and have exact quantum 1-query complexity is not bigger than $n^{2}2^{2^{n-1}(1+2^{2-k})+2n^{2}}$ for all $n\geq 3$ and $k\geq 2$.

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