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A Direct Reduction from the Polynomial to the Adversary Method

Author Aleksandrs Belovs

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Author Details

Aleksandrs Belovs
  • Faculty of Computing, University of Latvia, Riga, Latvia

Funding

  • Belovs, Aleksandrs: This research is supported by the Latvian Quantum Initiative under European Union Recovery and Resilience Facility project no. 2.3.1.1.i.0/1/22/I/CFLA/001. Part of this work has been supported by the ERDF project number 1.1.1.5/18/A/020 "Quantum algorithms: from complexity theory to experiment."

Acknowledgements

The author is thankful to Shalev Ben-David for the suggestion to apply this construction to the polynomial method, and to anonymous referees for their helpful comments on the previous versions of the paper.

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Aleksandrs Belovs. A Direct Reduction from the Polynomial to the Adversary Method. In 19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 310, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.TQC.2024.11

Abstract

The polynomial and the adversary methods are the two main tools for proving lower bounds on query complexity of quantum algorithms. Both methods have found a large number of applications, some problems more suitable for one method, some for the other.
It is known though that the adversary method, in its general negative-weighted version, is tight for bounded-error quantum algorithms, whereas the polynomial method is not. By the tightness of the former, for any polynomial lower bound, there ought to exist a corresponding adversary lower bound. However, direct reduction was not known.
In this paper, we give a simple and direct reduction from the polynomial method (in the form of a dual polynomial) to the adversary method. This shows that any lower bound in the form of a dual polynomial is actually an adversary lower bound of a specific form.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum query complexity
Keywords
  • Polynomials
  • Quantum Adversary Bound

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