dorsal/arxiv
View SchemaLimitations of Quantum Coset States for Graph Isomorphism
| Authors | Sean Hallgren, Martin Roetteler, Pranab Sen |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0511148 |
| URL | https://arxiv.org/abs/quant-ph/0511148 |
| Journal | J. ACM 57(6): 34 (2010) |
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, Russell, and Schulman 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 \Omega(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 highly entangled measurements seem hard to implement in general. 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 G is finite and satisfies a suitable property.
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"abstract": "It has been known for some time that graph isomorphism reduces to the hidden\nsubgroup problem (HSP). What is more, most exponential speedups in quantum\ncomputation are obtained by solving instances of the HSP. A common feature of\nthe resulting algorithms is the use of quantum coset states, which encode the\nhidden subgroup. An open question has been how hard it is to use these states\nto solve graph isomorphism. It was recently shown by Moore, Russell, and\nSchulman that only an exponentially small amount of information is available\nfrom one, or a pair of coset states. A potential source of power to exploit are\nentangled quantum measurements that act jointly on many states at once. We show\nthat entangled quantum measurements on at least \\Omega(n log n) coset states\nare necessary to get useful information for the case of graph isomorphism,\nmatching an information theoretic upper bound. This may be viewed as a negative\nresult because highly entangled measurements seem hard to implement in general.\nOur main theorem is very general and also rules out using joint measurements on\nfew coset states for some other groups, such as GL(n, F_{p^m}) and G^n where G\nis finite and satisfies a suitable property.",
"arxiv_id": "quant-ph/0511148",
"authors": [
"Sean Hallgren",
"Martin Roetteler",
"Pranab Sen"
],
"categories": [
"quant-ph",
"cs.ET"
],
"journal_ref": "J. ACM 57(6): 34 (2010)",
"title": "Limitations of Quantum Coset States for Graph Isomorphism",
"url": "https://arxiv.org/abs/quant-ph/0511148"
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