Partial Boolean Functions With Exact Quantum Query Complexity One

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 algorit...

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Bibliographic Details
Published in:Entropy (Basel, Switzerland) Vol. 23; no. 2; p. 189
Main Authors: Xu, Guoliang, Qiu, Daowen
Format: Journal Article
Language:English
Published: Switzerland MDPI AG 03.02.2021
MDPI
Subjects:
Algorithms
Boolean
Boolean algebra
Boolean functions
Codes
Complexity
Computation
Decision trees
partial Boolean function
quantum computation
Quantum computing
quantum query algorithm
quantum query complexity
Queries
Upper bounds
partial Boolean function
quantum query complexity
quantum computation
quantum query algorithm
ISSN:1099-4300, 1099-4300
Online Access:Get full text
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Summary: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 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 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.
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ISSN:1099-4300
1099-4300
DOI:10.3390/e23020189