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A Tight Lower Bound For Non-Coherent Index Erasure

Authors Nathan Lindzey, Ansis Rosmanis

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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 AsGet 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

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