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

Authors Aleksandrs Belovs, Ansis Rosmanis

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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

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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)


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
  • Adversary Bound
  • Dual Learning Graphs
  • Quantum Query Complexity
  • Representation Theory


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  1. Scott Aaronson and Andris Ambainis. Forrelation: A problem that optimally separates quantum from classical computing. In Proc. of 47th ACM STOC, pages 307-316, 2015. Google Scholar
  2. Scott Aaronson, Shalev Ben-David, and Robin Kothari. Separations in query complexity using cheat sheets. In Proc. of 48th ACM STOC, pages 863-876, 2016. Google Scholar
  3. Andris Ambainis. Quantum lower bounds by quantum arguments. Journal of Computer and System Sciences, 64(4):750-767, 2002. Google Scholar
  4. Aleksandrs Belovs. Span programs for functions with constant-sized 1-certificates. In Proc. of 44th ACM STOC, pages 77-84, 2012. Google Scholar
  5. Aleksandrs Belovs. Applications of the Adversary Method in Quantum Query Algorithms. PhD thesis, University of Latvia, 2014. Google Scholar
  6. Aleksandrs Belovs and Ansis Rosmanis. On the power of non-adaptive learning graphs. Computational Complexity, 23(2):323-354, 2014. Google Scholar
  7. Aleksandrs Belovs and Ansis Rosmanis. Adversary lower bounds for the collision and the set equality problems. Quantum Information &Computation, 18(3&4):198-222, 2018. Google Scholar
  8. Aleksandrs Belovs and Ansis Rosmanis. Quantum lower bounds for tripartite versions of the hidden shift and the set equality problems. Full version. Avaiable at arXiv:1712.10194, 2018. Google Scholar
  9. Aleksandrs Belovs and Robert Špalek. Adversary lower bound for the k-sum problem. In Proc. of 4th ACM ITCS, pages 323-328, 2013. Google Scholar
  10. Shalev Ben-David. A super-Grover separation between randomized and quantum query complexities, 2015. URL: http://arxiv.org/abs/1506.08106.
  11. Harry Buhrman and Ronald de Wolf. Complexity measures and decision tree complexity: a survey. Theoretical Computer Science, 288:21-43, 2002. Google Scholar
  12. Charles W. Curtis and Irving Reiner. Representation theory of finite groups and associative algebras. AMS, 1962. Google Scholar
  13. Mark Ettinger, Peter Høyer, and Emanuel Knill. The quantum query complexity of the hidden subgroup problem is polynomial. Information Processing Letters, 91(1):43-48, 2004. Google Scholar
  14. Aram W. Harrow, Cedric Yen-Yu Lin, and Ashley Montanaro. Sequential measurements, disturbance and property testing. In Proc. of 28th ACM-SIAM SODA, pages 1598-1611, 2017. Google Scholar
  15. Peter Høyer, Troy Lee, and Robert Špalek. Negative weights make adversaries stronger. In Proc. of 39th ACM STOC, pages 526-535, 2007. Google Scholar
  16. Gordon James and Adalbert Kerber. The Representation Theory of the Symmetric Group, volume 16 of Encyclopedia of Mathematics and its Applications. Addison-Wesley, 1981. Google Scholar
  17. Greg Kuperberg. A subexponential-time quantum algorithm for the dihedral hidden subgroup problem. SIAM Journal on Computing, 35:170-188, 2005. Google Scholar
  18. Troy Lee, Rajat Mittal, Ben W. Reichardt, Robert Špalek, and Mario Szegedy. Quantum query complexity of state conversion. In Proc. of 52nd IEEE FOCS, pages 344-353, 2011. Google Scholar
  19. Ashley Montanaro and Ronald de Wolf. A survey of quantum property testing. Theory of Computing Graduate Surveys, 7:1-81, 2016. Google Scholar
  20. Ben W. Reichardt. Span programs and quantum query complexity: The general adversary bound is nearly tight for every Boolean function. In Proc. of 50th IEEE FOCS, pages 544-551, 2009. Google Scholar
  21. Bruce E. Sagan. The symmetric group: representations, combinatorial algorithms, and symmetric functions, volume 203 of Graduate Texts in Mathematics. Springer, 2001. Google Scholar
  22. Jean-Pierre Serre. Linear Representations of Finite Groups, volume 42 of Graduate Texts in Mathematics. Springer, 1977. Google Scholar
  23. Yaoyun Shi. Quantum lower bounds for the collision and the element distinctness problems. In Proc. of 43th IEEE FOCS, pages 513-519, 2002. Google Scholar
  24. Robert Špalek and Mario Szegedy. All quantum adversary methods are equivalent. Theory of Computing, 2:1-18, 2006. Google Scholar
  25. Mark Zhandry. A note on the quantum collision and set equality problems. Quantum Information &Computation, 15(7&8):557-567, 2015. Google Scholar
  26. Shengyu Zhang. On the power of Ambainis lower bounds. Theoretical Computer Science, 339(2):241-256, 2005. Google Scholar
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