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

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Summary

Introduction

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

Preliminaries
A Characterization by the Linear System of Equations
Partial Boolean Functions Depending on All Bits
A Characterization by the Sum-of-Squares Representation
Partial Boolean Functions Depending on k Bits
Estimating the Number of Partial Boolean Functions Depending on k Bits
Conclusions
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