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Quantum Lower Bounds for Tripartite Versions of the Hidden Shift and the Set Equality Problems

Authors Aleksandrs Belovs, Ansis Rosmanis

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Author Details

Aleksandrs Belovs
  • Faculty of Computing, University of Latvia, Raina 19, Riga, Latvia
Ansis Rosmanis
  • Centre for Quantum Technologies, National University of Singapore, Block S15, 3 Science Drive 2, Singapore

Funding

  • Belovs, Aleksandrs: This research is partially supported by the ERDF project number 1.1.1.2/I/16/113. Part of this work was done while supported by the ERC Advanced Grant MQC.
  • Rosmanis, Ansis: This research is partially funded by the Singapore Ministry of Education and the National Research Foundation under grant R-710-000-012-135.

Cite As Get BibTex

Aleksandrs Belovs and Ansis Rosmanis. Quantum Lower Bounds for Tripartite Versions of the Hidden Shift and the Set Equality Problems. In 13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 111, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.TQC.2018.3

Abstract

In this paper, we study quantum query complexity of the following rather natural tripartite generalisations (in the spirit of the 3-sum problem) of the hidden shift and the set equality problems, which we call the 3-shift-sum and the 3-matching-sum problems.
The 3-shift-sum problem is as follows: given a table of 3 x n elements, is it possible to circularly shift its rows so that the sum of the elements in each column becomes zero? It is promised that, if this is not the case, then no 3 elements in the table sum up to zero. The 3-matching-sum problem is defined similarly, but it is allowed to arbitrarily permute elements within each row. For these problems, we prove lower bounds of Omega(n^{1/3}) and Omega(sqrt n), respectively. The second lower bound is tight.
The lower bounds are proven by a novel application of the dual learning graph framework and by using representation-theoretic tools from [Belovs, 2018].

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum query complexity
Keywords
  • Adversary Bound
  • Dual Learning Graphs
  • Quantum Query Complexity
  • Representation Theory

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