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Weak Fourier-Schur Sampling, the Hidden Subgroup Problem, and the Quantum Collision Problem

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STACS 2007 (STACS 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4393))

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Abstract

Schur duality decomposes many copies of a quantum state into subspaces labeled by partitions, a decomposition with applications throughout quantum information theory. Here we consider applying Schur duality to the problem of distinguishing coset states in the standard approach to the hidden subgroup problem. We observe that simply measuring the partition (a procedure we call weak Schur sampling) provides very little information about the hidden subgroup. Furthermore, we show that under quite general assumptions, even a combination of weak Fourier sampling and weak Schur sampling fails to identify the hidden subgroup. We also prove tight bounds on how many coset states are required to solve the hidden subgroup problem by weak Schur sampling, and we relate this question to a quantum version of the collision problem.

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Authors and Affiliations

  1. Institute for Quantum Information, California Institute of Technology, Pasadena, CA 91125, USA

    Andrew M. Childs

  2. Department of Computer Science, University of Bristol, Bristol, BS8 1UB, UK

    Aram W. Harrow

  3. School of Electrical Engineering and Computer Science, University of Central Florida, Orlando, FL 32816, USA

    Paweł Wocjan

Authors
  1. Andrew M. Childs
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  2. Aram W. Harrow
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  3. Paweł Wocjan
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Wolfgang Thomas Pascal Weil

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Childs, A.M., Harrow, A.W., Wocjan, P. (2007). Weak Fourier-Schur Sampling, the Hidden Subgroup Problem, and the Quantum Collision Problem. In: Thomas, W., Weil, P. (eds) STACS 2007. STACS 2007. Lecture Notes in Computer Science, vol 4393. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70918-3_51

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  • DOI: https://doi.org/10.1007/978-3-540-70918-3_51

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-70917-6

  • Online ISBN: 978-3-540-70918-3

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