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Abelian Hidden Subgroup Problem

  • Reference work entry
  • First Online:
Encyclopedia of Algorithms

  • 1360 Accesses

Years and Authors of Summarized Original Work

  • 1995; Kitaev

  • 2008; Mosca

Problem Definition

The Abelian hidden subgroup problem is the problem of finding generators for a subgroup K of an Abelian group G, where this subgroup is defined implicitly by a function f : G → X, for some finite set X. In particular, f has the property that f(v) = f(w) if and only if the cosets (we are assuming additive notation for the group operation here.) v + K and w + K are equal. In other words, f is constant on the cosets of the subgroup K and distinct on each coset.

It is assumed that the group G is finitely generated and that the elements of G and X have unique binary encodings. The binary assumption is only for convenience, but it is important to have unique encodings (e.g., in [22] Watrous uses a quantum state as the unique encoding of group elements). When using variables g and h (possibly with subscripts), multiplicative notation is used for the group operations. Variables x and y(possibly with...

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Recommended Reading

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Author information

Authors and Affiliations

  1. Canadian Institute for Advanced Research, Toronto, ON, Canada

    Michele Mosca

  2. Combinatorics and Optimization/Institute for Quantum Computing, University of Waterloo, Waterloo, ON, Canada

    Michele Mosca

  3. Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada

    Michele Mosca

Authors
  1. Michele Mosca
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Corresponding author

Correspondence to Michele Mosca .

Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL, USA

    Ming-Yang Kao

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Mosca, M. (2016). Abelian Hidden Subgroup Problem. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_1

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