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# Exponential Quantum Communication Reductions from Generalizations of the Boolean Hidden Matching Problem

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LIPIcs.TQC.2020.1.pdf
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## Acknowledgements

We would like to thank Ronald de Wolf for pointing out Ref. [Shi et al., 2012], and Makrand Sinha for useful discussions about the hypercontractive inequality.

## Cite As

João F. Doriguello and Ashley Montanaro. Exponential Quantum Communication Reductions from Generalizations of the Boolean Hidden Matching Problem. In 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 158, pp. 1:1-1:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.TQC.2020.1

## Abstract

In this work we revisit the Boolean Hidden Matching communication problem, which was the first communication problem in the one-way model to demonstrate an exponential classical-quantum communication separation. In this problem, Alice’s bits are matched into pairs according to a partition that Bob holds. These pairs are compressed using a Parity function and it is promised that the final bit-string is equal either to another bit-string Bob holds, or its complement. The problem is to decide which case is the correct one. Here we generalize the Boolean Hidden Matching problem by replacing the parity function with an arbitrary function f. Efficient communication protocols are presented depending on the sign-degree of f. If its sign-degree is less than or equal to 1, we show an efficient classical protocol. If its sign-degree is less than or equal to 2, we show an efficient quantum protocol. We then completely characterize the classical hardness of all symmetric functions f of sign-degree greater than or equal to 2, except for one family of specific cases. We also prove, via Fourier analysis, a classical lower bound for any function f whose pure high degree is greater than or equal to 2. Similarly, we prove, also via Fourier analysis, a quantum lower bound for any function f whose pure high degree is greater than or equal to 3. These results give a large family of new exponential classical-quantum communication separations.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Communication complexity
• Theory of computation → Quantum complexity theory
##### Keywords
• Communication Complexity
• Quantum Communication Complexity
• Boolean Hidden Matching Problem

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