An Application of the Deutsch-Jozsa Algorithm to Formal Languages and the Word Problem in Groups

  • Authors:

  • Michael Batty;Andrea Casaccino;Andrew J. Duncan;Sarah Rees;Simone Severini

  • Affiliations:

  • Department of Mathematics, University of Newcastle upon Tyne, United Kingdom;Information Engineering Department, University of Siena, Italy;Department of Mathematics, University of Newcastle upon Tyne, United Kingdom;Department of Mathematics, University of Newcastle upon Tyne, United Kingdom;Institute for Quantum Computing and Department of Combinatorics and Optimization, University of Waterloo, Canada

  • Venue:
  • Theory of Quantum Computation, Communication, and Cryptography
  • Year:
  • 2008

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Abstract

We adapt the Deutsch-Jozsa algorithm to the context of formal language theory. Specifically, we use the algorithm to distinguish between trivial and nontrivial words in groups given by finite presentations, under the promise that a word is of a certain type. This is done by extending the original algorithm to functions of arbitrary length binary output, with the introduction of a more general concept of parity. We provide examples in which properties of the algorithm allow to reduce the number of oracle queries with respect to the deterministic classical case. This has some consequences for the word problem in groups with a particular kind of presentation.