Lower Bounds on the Deterministic and Quantum Communication Complexity of Hamming-Distance Problems

2015 ◽  
Vol 7 (3) ◽  
pp. 1-10 ◽  
Author(s):  
Andris Ambainis ◽  
William Gasarch ◽  
Aravind Srinivasan ◽  
Andrey Utis
Keyword(s):  
Lower Bounds ◽  
Communication Complexity ◽  
Quantum Communication ◽  
Hamming Distance ◽  
Quantum Communication Complexity

On the orthogonal rank and impossibility of quantum round elimination

2017 ◽  
Vol 17 (1&2) ◽  
pp. 106-116
Author(s):  
Jop Briet ◽  
Jeroen Zuiddam
Keyword(s):  
Lower Bound ◽  
Communication Complexity ◽  
Quantum Communication ◽  
Hamming Distance ◽  
Cayley Graphs ◽  
Affirmative Answer ◽  
Complex Vector ◽  
Total Communication ◽  
Quantum Communication Complexity ◽  
The Cost

After Bob sends Alice a bit, she responds with a lengthy reply. At the cost of a factor of two in the total communication, Alice could just as well have given Bob her two possible replies at once without listening to him at all, and have him select which one applies. Motivated by a conjecture stating that this form of “round elimination” is impossible in exact quantum communication complexity, we study the orthogonal rank and a symmetric variant thereof for a certain family of Cayley graphs. The orthogonal rank of a graph is the smallest number d for which one can label each vertex with a nonzero d-dimensional complex vector such that adjacent vertices receive orthogonal vectors. We show an exp(n) lower bound on the orthogonal rank of the graph on {0, 1} n in which two strings are adjacent if they have Hamming distance at least n/2. In combination with previous work, this implies an affirmative answer to the above conjecture.

scholarly journals Generalizations of the distributed Deutsch–Jozsa promise problem

2015 ◽  
Vol 27 (3) ◽  
pp. 311-331 ◽  
Author(s):  
JOZEF GRUSKA ◽  
DAOWEN QIU ◽  
SHENGGEN ZHENG
Keyword(s):  
Communication Complexity ◽  
Quantum Communication ◽  
Hamming Distance ◽  
Finite Automata ◽  
Classical Communication ◽  
Deterministic Finite Automata ◽  
Exact Quantum ◽  
Quantum Communication Complexity

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.

scholarly journals Symmetrized persistency of Bell correlations for Dicke states and GHZ-based mixtures

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Marcin Wieśniak
Keyword(s):  
Communication Complexity ◽  
Quantum Communication ◽  
Bell Inequality ◽  
Bell Inequalities ◽  
Quantum Correlations ◽  
Complexity Reduction ◽  
Main Difficulty ◽  
Quantum Communication Complexity ◽  
Number Of Particles ◽  
Dicke States

AbstractQuantum correlations, in particular those, which enable to violate a Bell inequality, open a way to advantage in certain communication tasks. However, the main difficulty in harnessing quantumness is its fragility to, e.g, noise or loss of particles. We study the persistency of Bell correlations of GHZ based mixtures and Dicke states. For the former, we consider quantum communication complexity reduction (QCCR) scheme, and propose new Bell inequalities (BIs), which can be used in that scheme for higher persistency in the limit of large number of particles N. In case of Dicke states, we show that persistency can reach 0.482N, significantly more than reported in previous studies.

Exponential Separation for One-Way Quantum Communication Complexity, with Applications to Cryptography

2009 ◽  
Vol 38 (5) ◽  
pp. 1695-1708 ◽  
Author(s):  
Dmitry Gavinsky ◽  
Julia Kempe ◽  
Iordanis Kerenidis ◽  
Ran Raz ◽  
Ronald de Wolf
Keyword(s):  
Communication Complexity ◽  
Quantum Communication ◽  
Exponential Separation ◽  
Quantum Communication Complexity
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