Reconstructing strings from substrings with quantum queries

  • Authors:

  • Richard Cleve;Kazuo Iwama;François Le Gall;Harumichi Nishimura;Seiichiro Tani;Junichi Teruyama;Shigeru Yamashita

  • Affiliations:

  • Institute for Quantum Computing and School of Computer Science, University of Waterloo, Canada,Perimeter Institute for Theoretical Physics, Canada;School of Informatics, Kyoto University, Japan;Department of Computer Science, The University of Tokyo, Japan;School of Information Science, Nagoya University, Japan;NTT Communication Science Laboratories, NTT Corporation, Atsugi, Japan;School of Informatics, Kyoto University, Japan;College of Information Science and Engineering, Ritsumeikan University, Japan

  • Venue:
  • SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
  • Year:
  • 2012

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Abstract

This paper investigates the number of quantum queries made to solve the problem of reconstructing an unknown string from its substrings in a certain query model. More concretely, the goal of the problem is to identify an unknown string S by making queries of the following form: "Is s a substring of S?", where s is a query string over the given alphabet. The number of queries required to identify the string S is the query complexity of this problem. First we show a quantum algorithm that exactly identifies the string S with at most $\frac{3}{4}N + o(N)$ queries, where N is the length of S. This contrasts sharply with the classical query complexity N. Our algorithm uses Skiena and Sundaram's classical algorithm and the Grover search as subroutines. To make them effectively work, we develop another subroutine that finds a string appearing only once in S, which may have an independent interest. We also prove two lower bounds. The first one is a general lower bound of $\Omega(\frac{N}{\log^2{N}})$, which means we cannot achieve a query complexity of O(N1−ε) for any constant ε. The other one claims that if we cannot use queries of length roughly between logN and 3 logN, then we cannot achieve a query complexity of any sublinear function in N.