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Quantum Query-To-Communication Simulation Needs a Logarithmic Overhead

Authors Sourav Chakraborty, Arkadev Chattopadhyay, Nikhil S. Mande, Manaswi Paraashar

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Sourav Chakraborty
  • Indian Statistical Institute, Kolkata, India
Arkadev Chattopadhyay
  • Tata Institute of Fundamental Research, Mumbai, India
Nikhil S. Mande
  • Georgetown University, Washington, D.C., USA
Manaswi Paraashar
  • Indian Statistical Institute, Kolkata, India

Acknowledgements

We thank Ronald de Wolf and Srinivasan Arunachalam for various discussions on this problem. They provided us with a number of important pointers that were essential for our understanding of the problem and its importance. We thank Justin Thaler for referring us to Claim 31. We thank the anonymous CCC 2020 reviewers for invaluable comments. In particular, we thank one of the reviewers for pointing out the contrasts between the classical query model and the quantum query model that our results imply (Remark 5).

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Sourav Chakraborty, Arkadev Chattopadhyay, Nikhil S. Mande, and Manaswi Paraashar. Quantum Query-To-Communication Simulation Needs a Logarithmic Overhead. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 32:1-32:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.CCC.2020.32

Abstract

Buhrman, Cleve and Wigderson (STOC'98) observed that for every Boolean function f:{-1,1}ⁿ → {-1,1} and •:{-1,1}² → {-1,1} the two-party bounded-error quantum communication complexity of (f ∘ •) is O(Q(f) log n), where Q(f) is the bounded-error quantum query complexity of f. Note that the bounded-error randomized communication complexity of (f ∘ •) is bounded by O(R(f)), where R(f) denotes the bounded-error randomized query complexity of f. Thus, the BCW simulation has an extra O(log n) factor appearing that is absent in classical simulation. A natural question is if this factor can be avoided. Razborov (IZV MATH'03) showed that the bounded-error quantum communication complexity of Set-Disjointness is Ω(√n). The BCW simulation yields an upper bound of O(√n log n). Høyer and de Wolf (STACS'02) showed that this can be reduced to c^(log^* n) for some constant c, and subsequently Aaronson and Ambainis (FOCS'03) showed that this factor can be made a constant. That is, the quantum communication complexity of the Set-Disjointness function (which is NOR_n ∘ ∧) is O(Q(NOR_n)). Perhaps somewhat surprisingly, we show that when • = ⊕, then the extra log n factor in the BCW simulation is unavoidable. In other words, we exhibit a total function F:{-1,1}ⁿ → {-1,1} such that Q^{cc}(F ∘ ⊕) = Θ(Q(F) log n). To the best of our knowledge, it was not even known prior to this work whether there existed a total function F and 2-bit function •, such that Q^{cc}(F ∘ •) = ω(Q(F)).

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum query complexity
  • Theory of computation → Quantum communication complexity
Keywords
  • Quantum query complexity
  • quantum communication complexity
  • approximate degree
  • approximate spectral norm

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