### 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 _{pm}) and G^{n} where G is finite and satisfies a suitable property.

Original language | English (US) |
---|---|

Article number | 34 |

Journal | Journal of the ACM |

Volume | 57 |

Issue number | 6 |

DOIs | |

State | Published - Oct 1 2010 |

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### All Science Journal Classification (ASJC) codes

- Software
- Control and Systems Engineering
- Information Systems
- Hardware and Architecture
- Artificial Intelligence

### Cite this

*Journal of the ACM*,

*57*(6), [34]. https://doi.org/10.1145/1857914.1857918

}

*Journal of the ACM*, vol. 57, no. 6, 34. https://doi.org/10.1145/1857914.1857918

**Limitations of quantum coset states for graph isomorphism.** / Hallgren, Sean; Moore, Cristopher; Rötteler, Martin; Russell, Alexander; Sen, Pranab.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Limitations of quantum coset states for graph isomorphism

AU - Hallgren, Sean

AU - Moore, Cristopher

AU - Rötteler, Martin

AU - Russell, Alexander

AU - Sen, Pranab

PY - 2010/10/1

Y1 - 2010/10/1

N2 - 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 pm) and Gn where G is finite and satisfies a suitable property.

AB - 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 pm) and Gn where G is finite and satisfies a suitable property.

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UR - http://www.scopus.com/inward/citedby.url?scp=78649239030&partnerID=8YFLogxK

U2 - 10.1145/1857914.1857918

DO - 10.1145/1857914.1857918

M3 - Article

VL - 57

JO - Journal of the ACM

JF - Journal of the ACM

SN - 0004-5411

IS - 6

M1 - 34

ER -