Partial Boolean functions with exact quantum 1-query complexity
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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^22^2^n-1(1+2^2-k)+2n^2 for all n≥ 3 and k≥ 2.
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