### Limitations of quantum coset states for graph isomorphismLimitations of quantum coset states for graph isomorphism

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 Author Hallgren, Sean ♦ Moore, Cristopher ♦ Rtteler, Martin ♦ Russell, Alexander ♦ Sen, Pranab Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©2010 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Subject Keyword Graph isomorphism ♦ Hidden subgroup problem ♦ Quantum algorithms ♦ Quantum computation 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 $Ω(\textit{n}$ log $\textit{n})$ 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 $G^{n}$ where $\textit{G}$ is finite and satisfies a suitable property. ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 2010-11-05 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 57 Issue Number 6 Page Count 33 Starting Page 1 Ending Page 33

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Source: ACM Digital Library