Abstract
We provide two sufficient and necessary conditions to characterize any n-bit partial Boolean function with exact quantum query complexity 1. Using the first characterization, we present all n-bit partial Boolean functions that depend on n bits and can be computed exactly by a 1-query quantum algorithm. 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 can be computed exactly by a 1-query quantum algorithm, then is non-positive. In addition, we show that the number of all n-bit partial Boolean functions that depend on k bits and can be computed exactly by a 1-query quantum algorithm is not bigger than an upper bound depending on n and k. Most importantly, the upper bound is far less than the number of all n-bit partial Boolean functions for all efficiently big n.
Highlights
Boolean Functions with ExactIn the field of theoretical computer science, computational complexity aims to measure “how much" computation is necessary and sufficient to finish some certain computational tasks
Given any n-bit partial Boolean function f, if some equations (By basic linear algebra, n + 1 equations are enough in the best case) lead to an empty solution of the linear system, by Theorem 1 these equations are enough to prove that the exact quantum query complexity of f is bigger than 1
The notation Nj (n, k ) is used to denote the number of n-bit partial Boolean functions which depend on k bits and can be computed exactly by a j-query quantum algorithm
Summary
In the field of theoretical computer science, computational complexity aims to measure “how much" computation is necessary and sufficient to finish some certain computational tasks. Given an input x ∈ D ⊆ {0, 1}n that can only be accessed through a black box by querying some bit xi of the input, the quantum query model computes an n-bit partial Boolean function f : D → {0, 1} exactly (or with bounded-error) [1]. The exact (or boundederror) quantum query complexity of a Boolean function denotes the number of queries of an optimal quantum decision tree that computes the Boolean function exactly (or with bounded-error) [1]. The exact 1-query quantum model for all partial Boolean functions is expected to be investigated further. The number of partial Boolean functions with exact quantum query complexity 1 shows the power and advantage of the exact 1-query quantum model. For the sake of brevity and readability, all proofs of lemmas in this paper are showed in Appendices A–D
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