research-article

Limitations of quantum coset states for graph isomorphism

Authors:

Cristopher Moore,

Martin Rötteler,

Alexander Russell,

Pranab SenAuthors Info & Claims

Journal of the ACM (JACM), Volume 57, Issue 6

Article No.: 34, Pages 1 - 33

https://doi.org/10.1145/1857914.1857918

Published: 05 November 2010 Publication History

Abstract

It has been known for some time that graph isomorphism reduces to the hidden subgroup problem (HSP). What is more, most exponential speedups in quantum computation are obtained by solving instances of the HSP. A common feature of the resulting algorithms is the use of quantum coset states, which encode the hidden subgroup. An open question has been how hard it is to use these states to solve graph isomorphism. It was recently shown by Moore et al. [2005] that only an exponentially small amount of information is available from one, or a pair of coset states. A potential source of power to exploit are entangled quantum measurements that act jointly on many states at once.

We show that entangled quantum measurements on at least Ω(n log n) coset states are necessary to get useful information for the case of graph isomorphism, matching an information theoretic upper bound. This may be viewed as a negative result because in general it seems hard to implement a given highly entangled measurement. Our main theorem is very general and also rules out using joint measurements on few coset states for some other groups, such as GL(n,F_p^m) and Gⁿ where G is finite and satisfies a suitable property.

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Index Terms

Limitations of quantum coset states for graph isomorphism
1. Theory of computation
  1. Logic
  2. Models of computation

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cover image Journal of the ACM

Journal of the ACM Volume 57, Issue 6

October 2010

151 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/1857914

Issue’s Table of Contents

Copyright © 2010 ACM.

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Publication History

Published: 05 November 2010

Accepted: 01 March 2010

Revised: 01 March 2010

Received: 01 October 2008

Published in JACM Volume 57, Issue 6

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Hhan MYamakawa TYun A(2024)Quantum Complexity for Discrete Logarithms and Related ProblemsAdvances in Cryptology – CRYPTO 202410.1007/978-3-031-68391-6_1(3-36)Online publication date: 17-Aug-2024
https://doi.org/10.1007/978-3-031-68391-6_1
Bläser MChen ZDuong DJoux ANguyen TPlantard TQiao YSusilo WTang G(2024)On Digital Signatures Based on Group Actions: QROM Security and Ring SignaturesPost-Quantum Cryptography10.1007/978-3-031-62743-9_8(227-261)Online publication date: 12-Jun-2024
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