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Generalizations of the distributed Deutsch–Jozsa promise problem

Published online by Cambridge University Press:  06 May 2015

JOZEF GRUSKA ,
DAOWEN QIU  and
SHENGGEN ZHENG
JOZEF GRUSKA
Affiliation:
Faculty of Informatics, Masaryk University, Brno 60200, Czech Republic
DAOWEN QIU
Affiliation:
Department of Computer Science, Sun Yat-sen University, Guangzhou 510006, China Email: zhengshenggen@gmail.com
SHENGGEN ZHENG*
Affiliation:
Faculty of Informatics, Masaryk University, Brno 60200, Czech Republic Department of Computer Science, Sun Yat-sen University, Guangzhou 510006, China Email: zhengshenggen@gmail.com
*
‡Corresponding author. Zheng was supported by the Employment of Newly Graduated Doctors of Science for Scientific Excellence project/grant (CZ.1.07./2.3.00/30.0009) of Czech Republic.
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Abstract

In the distributed Deutsch–Jozsa promise problem, two parties are to determine whether their respective strings x, y ∈ {0,1}n are at the Hamming distanceH(x, y) = 0 or H(x, y) = $\frac{n}{2}$. Buhrman et al. (STOC' 98) proved that the exact quantum communication complexity of this problem is O(log n) while the deterministic communication complexity is Ω(n). This was the first impressive (exponential) gap between quantum and classical communication complexity. In this paper, we generalize the above distributed Deutsch–Jozsa promise problem to determine, for any fixed $\frac{n}{2}$ ⩽ k ⩽ n, whether H(x, y) = 0 or H(x, y) = k, and show that an exponential gap between exact quantum and deterministic communication complexity still holds if k is an even such that $\frac{1}{2}$n ⩽ k < (1 − λ)n, where 0 < λ < $\frac{1}{2}$ is given. We also deal with a promise version of the well-known disjointness problem and show also that for this promise problem there exists an exponential gap between quantum (and also probabilistic) communication complexity and deterministic communication complexity of the promise version of such a disjointness problem. Finally, some applications to quantum, probabilistic and deterministic finite automata of the results obtained are demonstrated.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 27 , Issue 3 , March 2017 , pp. 311 - 331
Copyright
Copyright © Cambridge University Press 2015 

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Footnotes

† Qiu was partially supported by the National Natural Science Foundation of China (Nos. 61272058, 61073054).

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