On the impossibility of a quantum sieve algorithm for graph isomorphism: unconditional results

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 Ω(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^{Ω(\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})}.

An earlier preprint, quant-ph/0609138, gave versions of these results which were conditional on a group-theoretic conjecture. This version provides unconditional results

References in corpus (5)

• From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups
• For Distinguishing Conjugate Hidden Subgroups, the Pretty Good Measurement is as Good as it Gets
• Upper bound on the characters of the symmetric groups for balanced Young diagrams and a generalized Frobenius formula
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Cited by in corpus (4)

• Upper bound on the characters of the symmetric groups for balanced Young diagrams and a generalized Frobenius formula
• Superpolynomial speedups based on almost any quantum circuit
• How a Clebsch-Gordan Transform Helps to Solve the Heisenberg Hidden Subgroup Problem
• Quantum Algorithms Using the Curvelet Transform