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On Query-To-Communication Lifting for Adversary Bounds

Authors Anurag Anshu, Shalev Ben-David, Srijita Kundu

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ACM Subject Classification
  • Theory of computation → Communication complexity
  • Theory of computation → Quantum complexity theory
Keywords
  • Quantum computing
  • query complexity
  • communication complexity
  • lifting theorems
  • adversary method

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Abstract

We investigate query-to-communication lifting theorems for models related to the quantum adversary bounds. Our results are as follows:  
1) We show that the classical adversary bound lifts to a lower bound on randomized communication complexity with a constant-sized gadget. We also show that the classical adversary bound is a strictly stronger lower bound technique than the previously-lifted measure known as critical block sensitivity, making our lifting theorem one of the strongest lifting theorems for randomized communication complexity using a constant-sized gadget. 
2) Turning to quantum models, we show a connection between lifting theorems for quantum adversary bounds and secure 2-party quantum computation in a certain "honest-but-curious" model. Under the assumption that such secure 2-party computation is impossible, we show that a simplified version of the positive-weight adversary bound lifts to a quantum communication lower bound using a constant-sized gadget. We also give an unconditional lifting theorem which lower bounds bounded-round quantum communication protocols. 
3) Finally, we give some new results in query complexity. We show that the classical adversary and the positive-weight quantum adversary are quadratically related. We also show that the positive-weight quantum adversary is never larger than the square of the approximate degree. Both relations hold even for partial functions.

Cite As Get BibTex

Anurag Anshu, Shalev Ben-David, and Srijita Kundu. On Query-To-Communication Lifting for Adversary Bounds. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 30:1-30:39, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.CCC.2021.30

Author Details

Anurag Anshu
  • EECS & Challenge Institute for Quantum Computation, University of California, Berkeley, CA, USA
  • Simons Institute for the Theory of Computing, Berkeley, CA, USA
Shalev Ben-David
  • University of Waterloo, Canada
Srijita Kundu
  • Centre for Quantum Technologies, National University of Singapore, Singapore

Funding

  • Anshu, Anurag: Supported by the NSF QLCI program through grant number OMA-2016245. Part of this work was done when A.A. was affiliated to the Department of Combinatorics & Optimization and Institute for Quantum Computing, University of Waterloo, as well as to the Perimeter Institute for Theoretical Physics.
  • Ben-David, Shalev: Supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), DGECR-2019-00027 and RGPIN-2019-04804.
  • Kundu, Srijita: Supported by the National Research Foundation, including under NRF RF Award No. NRF-NRFF2013-13, the Prime Minister’s Office, Singapore; the Ministry of Education, Singapore, under the Research Centres of Excellence program and by Grant No. MOE2012-T3-1-009; and in part by the NRF2017-NRF-ANR004 VanQuTe Grant. Part of this work was done when S.K. was visiting the Institute of Quantum Computing, University of Waterloo.

Acknowledgements

We thank Rahul Jain and Dave Touchette for helpful discussions related to the QICZ(G) > 0 conjecture. We thank Robin Kothari for helpful discussions related to the adversary bounds. We thank Anne Broadbent for helpful discussions related to quantum secure 2-party computation. We thank Mika Göös for helpful discussions regarding critical block sensitivity and its lifting theorem. We thank Jevg{ē}nijs Vihrovs and the other authors of [Ambainis et al., 2018] for helpful discussions regarding the classical adversary method, and particularly Krišjānis Prūsis for the proof of Lemma 28.

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