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Quantum Complexities of Ordered Searching, Sorting, and Element Distinctness

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Automata, Languages and Programming (ICALP 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2076))

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Abstract

We consider the quantum complexities of the following three problems: searching an ordered list, sorting an un-ordered list, and deciding whether the numbers in a list are all distinct. Letting N be the number of elements in the input list, we prove a lower bound of 1/π(ln(N) - 1) accesses to the list elements for ordered searching, a lower bound of Ω(N logN) binary comparisons for sorting, and a lower bound of Ω(√N logN) binary comparisons for element distinctness. The previously best known lower bounds are 1/12 log2(N) - O(1) due to Ambainis, Ω(N), and Ω(√N), respectively. Our proofs are based on a weighted all-pairs inner product argument.

In addition to our lower bound results, we give a quantum algorithm for ordered searching using roughly 0:631 log2(N) oracle accesses. Our algorithm uses a quantum routine for traversing through a binary search tree faster than classically, and it is of a nature very different from a faster algorithm due to Farhi, Goldstone, Gutmann, and Sipser.

Research supported by the EU fifth framework program QAIP, IST-1999-11234, and the National Science Foundation under grant CCR-9820855.

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Author information

Authors and Affiliations

  1. Dept. of Comp. Sci., University of Calgary, Alberta, Canada, T2N 1N4

    Peter Høyer

  2. Dept. of Comp. Sci., University of Aarhus, DK-8000, Århus C, Denmark

    Jan Neerbek

  3. Dept. of Comp. Sci., Princeton University, Princeton, NJ, 08544, USA

    Yaoyun Shi

Authors
  1. Peter Høyer
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  2. Jan Neerbek
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  3. Yaoyun Shi
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Editor information

Editors and Affiliations

  1. Departament de Llenguatges i Sistemes Informátics, Univ. Politècnica de Catalunya, C/Jordi Girona Salgado 1-3, 08034, Barcelona, Spain

    Fernando Orejas

  2. Computer Technology Institute (CTI), University of Patras, 61 Riga Feraiou street, 26221, Patras, Greece

    Paul G. Spirakis

  3. Institute of Information and Computing Sciences, Utrecht University, Padualaan 14, 3584, CH, Utrecht, The Netherlands

    Jan van Leeuwen

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© 2001 Springer-Verlag Berlin Heidelberg

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Høyer, P., Neerbek, J., Shi, Y. (2001). Quantum Complexities of Ordered Searching, Sorting, and Element Distinctness. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds) Automata, Languages and Programming. ICALP 2001. Lecture Notes in Computer Science, vol 2076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48224-5_29

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  • DOI: https://doi.org/10.1007/3-540-48224-5_29

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42287-7

  • Online ISBN: 978-3-540-48224-6

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