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Theory of Quantum Computation, Communication, and Cryptography

Volume 5106 of the series Lecture Notes in Computer Science pp 57-69

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

  • Michael BattyAffiliated withDepartment of Mathematics, University of Newcastle upon Tyne
  • , Andrea CasaccinoAffiliated withInformation Engineering Department, University of Siena
  • , Andrew J. DuncanAffiliated withDepartment of Mathematics, University of Newcastle upon Tyne
  • , Sarah ReesAffiliated withDepartment of Mathematics, University of Newcastle upon Tyne
  • , Simone SeveriniAffiliated withInstitute for Quantum Computing and Department of Combinatorics and Optimization, University of Waterloo

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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.