Document Open Access Logo

A Tight Lower Bound For Non-Coherent Index Erasure

Authors Nathan Lindzey, Ansis Rosmanis

PDF
Thumbnail PDF

File

LIPIcs.ITCS.2020.59.pdf
  • Filesize: 0.7 MB
  • 37 pages

Document Identifiers

Author Details

Nathan Lindzey
  • Department of Computer Science, University of Colorado at Boulder, USA
Ansis Rosmanis
  • Graduate School of Mathematics, Nagoya University, Japan

Funding

  • Rosmanis, Ansis: The author is a JSPS International Research Fellow supported by the JSPS KAKENHI Grant Number JP19F19079. Part of this work was done while he was at the Centre for Quantum Technologies at the National University of Singapore supported by the Singapore Ministry of Education and the National Research Foundation under grant R-710-000-012-135.

Acknowledgements

AR would like to thank Aleksandrs Belovs and Jérémie Roland for insightful discussions. NL thanks Chris Godsil for useful discussions.

Cite As Get BibTex

Nathan Lindzey and Ansis Rosmanis. A Tight Lower Bound For Non-Coherent Index Erasure. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 59:1-59:37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ITCS.2020.59

Abstract

The index erasure problem is a quantum state generation problem that asks a quantum computer to prepare a uniform superposition over the image of an injective function given by an oracle. We prove a tight Ω(√n) lower bound on the quantum query complexity of the non-coherent case of the problem, where, in addition to preparing the required superposition, the algorithm is allowed to leave the ancillary memory in an arbitrary function-dependent state. This resolves an open question of Ambainis, Magnin, Roetteler, and Roland (CCC 2011), who gave a tight bound for the coherent case, the case where the ancillary memory must return to its initial state.
To prove our main result, we first extend the so-called automorphism principle (Høyer et al. STOC 2007) to the general adversary method for state conversion problems (Lee et al. STOC 2011), which allows one to exploit the symmetries of these problems to lower bound their quantum query complexity. Using this method, we establish a strong connection between the quantum query complexity of non-coherent symmetric state generation problems and the well-known Krein parameters of association schemes. Krein parameters are usually hard to determine, nevertheless, we give a novel way of computing certain Krein parameters of a commutative association scheme defined over partial permutations. We believe the study of this association scheme may also be of independent interest.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum query complexity
  • Mathematics of computing → Combinatorics
Keywords
  • General Adversary Method
  • Quantum Query Complexity
  • Association Schemes
  • Krein Parameters
  • Representation Theory

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. A. Ambainis. Understanding quantum algorithms via query complexity. In Proceedings of the 2018 International Congress of Mathematicians, volume 3, pages 3249-3270, 2018. Google Scholar
  2. A. Ambainis, L. Magnin, M. Roetteler, and J. Roland. Symmetry-Assisted Adversaries for Quantum State Generation. In 2011 IEEE 26th Annual Conference on Computational Complexity, pages 167-177, June 2011. URL: https://doi.org/10.1109/CCC.2011.24.
  3. E. Bannai and T. Ito. Algebraic Combinatorics I: Association Schemes. Mathematics lecture note series. Benjamin/Cummings Pub. Co., 1984. Google Scholar
  4. A. Belovs and A. Rosmanis. Adversary lower bounds for the collision and the set equality problems. Quantum Information & Computation, 18:200-224, 2018. Google Scholar
  5. T. Ceccherini-Silberstein, F. Scarabotti, and F. Tolli. Harmonic Analysis on Finite Groups: Representation Theory, Gelfand Pairs and Markov Chains. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2008. Google Scholar
  6. P. Diaconis. Group Representations in Probability and Statistics. Institute of Mathematical Statistics Lecture Notes - Monograph Series, 11. Institute of Mathematical Statistics, Hayward, CA, 1988. Google Scholar
  7. D. Ellis, Y. Filmus, and E. Friedgut. LOW-DEGREE BOOLEAN FUNCTIONS ON S_n , WITH AN APPLICATION TO ISOPERIMETRY. Forum of Mathematics, Sigma, 5:e23, 2017. URL: https://doi.org/10.1017/fms.2017.24.
  8. D. Ellis, E. Friedgut, and H. Pilpel. Intersecting families of permutations. J. Amer. Math. Soc., 24:649-682, 2011. Google Scholar
  9. C. Godsil. Notes on Association Schemes, June 2010. Google Scholar
  10. C. Godsil and K. Meagher. Erdos-Ko-Rado Theorems: Algebraic Approaches. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2015. Google Scholar
  11. C. Godsil and G. Royle. Algebraic Graph Theory, volume 207 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2001. URL: https://doi.org/10.1007/978-1-4613-0163-9.
  12. A.S. Greenhalgh. Random Walks on Groups with Subgroup Invariance Properties. Technical report, Stanford University, Department of Statistics, April 1989. Google Scholar
  13. P. Høyer, T. Lee, and R. Špalek. Negative Weights Make Adversaries Stronger. In Proceedings of the Thirty-ninth Annual ACM Symposium on Theory of Computing, STOC '07, pages 526-535, New York, NY, USA, 2007. ACM. URL: https://doi.org/10.1145/1250790.1250867.
  14. G.D. James and A. Kerber. The Representation Theory of the Symmetric Group. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1984. Google Scholar
  15. T. Lee, R. Mittal, B.W. Reichardt, R. Špalek, and M. Szegedy. Quantum Query Complexity of State Conversion. In Proceedings of the 2011 IEEE 52Nd Annual Symposium on Foundations of Computer Science, FOCS '11, pages 344-353, Washington, DC, USA, 2011. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS.2011.75.
  16. G. Midrijānis. A Polynomial Quantum Query Lower Bound for the Set Equality Problem. In Automata, Languages and Programming, pages 996-1005, Berlin, Heidelberg, 2004. Springer Berlin Heidelberg. Google Scholar
  17. A. Montanaro. Quantum algorithms: An overview. npj Quantum Information, 2, November 2015. URL: https://doi.org/10.1038/npjqi.2015.23.
  18. M.A. Nielsen and I.L. Chuang. Quantum Computation and Quantum Information. Cambridge Series on Information and the Natural Sciences. Cambridge University Press, 2000. Google Scholar
  19. R. O'Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014. Google Scholar
  20. A. Rosmanis. Quantum Adversary Lower Bound for Element Distinctness with Small Range. Chicago Journal of Theoretical Computer Science, 2014(4), July 2014. Google Scholar
  21. A. Rosmanis and A. Belovs. On Adversary Lower Bounds for the Collision and the Set Equality Problems, 2013. Available at https://arxiv.org/abs/1310.5185v1 [quant-ph].
  22. B. Sagan. The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. Graduate Texts in Mathematics. Springer New York, 2001. Google Scholar
  23. Y. Shi. Quantum Lower Bounds for the Collision and the Element Distinctness Problems. In Proceedings of the 43rd Symposium on Foundations of Computer Science, FOCS '02, pages 513-519, Washington, DC, USA, 2002. IEEE Computer Society. Google Scholar
  24. R.P. Stanley. Enumerative Combinatorics: Volume 2. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2001. Google Scholar
  25. E. Strahov. Generalized characters of the symmetric group. Advances in Mathematics, 212(1):109-142, 2007. URL: https://doi.org/10.1016/j.aim.2006.09.017.
  26. M. Zhandry. A Note on the Quantum Collision and Set Equality Problems. Quantum Info. Comput., 15(7-8):557-567, May 2015. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact