License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.TQC.2018.3
URN: urn:nbn:de:0030-drops-92501
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9250/
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Belovs, Aleksandrs ; Rosmanis, Ansis

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

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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].

BibTeX - Entry

@InProceedings{belovs_et_al:LIPIcs:2018:9250,
  author =	{Aleksandrs Belovs and Ansis Rosmanis},
  title =	{{Quantum Lower Bounds for Tripartite Versions of the Hidden Shift and the Set Equality Problems}},
  booktitle =	{13th Conference on the Theory of Quantum Computation,  Communication and Cryptography (TQC 2018)},
  pages =	{3:1--3:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-080-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{111},
  editor =	{Stacey Jeffery},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9250},
  URN =		{urn:nbn:de:0030-drops-92501},
  doi =		{10.4230/LIPIcs.TQC.2018.3},
  annote =	{Keywords: Adversary Bound, Dual Learning Graphs, Quantum Query Complexity, Representation Theory}
}

Keywords: Adversary Bound, Dual Learning Graphs, Quantum Query Complexity, Representation Theory
Collection: 13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018)
Issue Date: 2018
Date of publication: 16.07.2018

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