Partial Boolean functions with exact quantum 1-query complexity

TL;DR

This paper characterizes and constructs partial Boolean functions with exact quantum 1-query complexity, providing conditions, classifications, and bounds on their number based on the number of dependent bits.

Contribution

It introduces two necessary and sufficient conditions for such functions, classifies all functions depending on all bits, and establishes an upper bound on their quantity.

Findings

Characterization of all n-bit partial Boolean functions with exact quantum 1-query complexity.

Construction of a function mapping partial Boolean functions to integers with specific properties.

Upper bound on the number of n-bit partial Boolean functions depending on k bits with the given complexity.

Abstract

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 $k\geq 2$.