dorsal/arxiv
View SchemaOn the impossibility of a quantum sieve algorithm for graph isomorphism: unconditional results
| Authors | Cristopher Moore, Alexander Russell, Piotr Sniady |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0612089 |
| URL | https://arxiv.org/abs/quant-ph/0612089 |
| DOI | 10.1145/1250790.1250868 |
| Journal | In: STOC '07: Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pages 536-545, New York, NY, USA, 2007. ACM Press |
Abstract
It is known that any quantum algorithm for Graph Isomorphism that works within the framework of the hidden subgroup problem (HSP) must perform highly entangled measurements across \Omega(n \log n) coset states. One of the only known models for how such a measurement could be carried out efficiently is Kuperberg's algorithm for the HSP in the dihedral group, in which quantum states are adaptively combined and measured according to the decomposition of tensor products into irreducible representations. This ``quantum sieve'' starts with coset states, and works its way down towards representations whose probabilities differ depending on, for example, whether the hidden subgroup is trivial or nontrivial. In this paper we show that no such approach can produce a polynomial-time quantum algorithm for Graph Isomorphism. Specifically, we consider the natural reduction of Graph Isomorphism to the HSP over the the wreath product S_n\wr Z_2. Using a recently proved bound on the irreducible characters of S_n, we show that no algorithm in this family can solve Graph Isomorphism in less than e^{\Omega(\sqrt{n})} time, no matter what adaptive rule it uses to select and combine quantum states. In particular, algorithms of this type can offer essentially no improvement over the best known classical algorithms, which run in time e^{O(\sqrt{n \log n})}.
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"abstract": "It is known that any quantum algorithm for Graph Isomorphism that works\nwithin the framework of the hidden subgroup problem (HSP) must perform highly\nentangled measurements across \\Omega(n \\log n) coset states. One of the only\nknown models for how such a measurement could be carried out efficiently is\nKuperberg\u0027s algorithm for the HSP in the dihedral group, in which quantum\nstates are adaptively combined and measured according to the decomposition of\ntensor products into irreducible representations. This ``quantum sieve\u0027\u0027 starts\nwith coset states, and works its way down towards representations whose\nprobabilities differ depending on, for example, whether the hidden subgroup is\ntrivial or nontrivial.\n In this paper we show that no such approach can produce a polynomial-time\nquantum algorithm for Graph Isomorphism. Specifically, we consider the natural\nreduction of Graph Isomorphism to the HSP over the the wreath product S_n\\wr\nZ_2. Using a recently proved bound on the irreducible characters of S_n, we\nshow that no algorithm in this family can solve Graph Isomorphism in less than\ne^{\\Omega(\\sqrt{n})} time, no matter what adaptive rule it uses to select and\ncombine quantum states. In particular, algorithms of this type can offer\nessentially no improvement over the best known classical algorithms, which run\nin time e^{O(\\sqrt{n \\log n})}.",
"arxiv_id": "quant-ph/0612089",
"authors": [
"Cristopher Moore",
"Alexander Russell",
"Piotr Sniady"
],
"categories": [
"quant-ph",
"math.RT"
],
"doi": "10.1145/1250790.1250868",
"journal_ref": "In: STOC \u002707: Proceedings of the thirty-ninth annual ACM symposium\n on Theory of computing, pages 536-545, New York, NY, USA, 2007. ACM Press",
"title": "On the impossibility of a quantum sieve algorithm for graph isomorphism: unconditional results",
"url": "https://arxiv.org/abs/quant-ph/0612089"
},
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