On the impossibility of a quantum sieve algorithm for graph isomorphism

Abstract

It is known that any algorithm for Graph Isomorphism thatworks within the framework of the hidden subgroup problem (HSP) must performhighly entangled measurements across Ω(n log n) coset states. One ofthe only known models for how such a measurement could be carried outefficiently is Kuperberg's algorithm for the HSP in the dihedral group, in whichquantum states are adaptively combined and measured according to thedecomposition of tensor products into irreducible representations. This quantum sieve starts with coset states, and works its way down towardsrepresentations whose probabilities differ depending on, for example, whetherthe hidden subgroup is trivial or nontrivial.In this paper we show that no such approach can produce a polynomial-time algorithm for Graph Isomorphism. Specifically, we consider the natural reduction of Graph Isomorphism to the HSP over the the wreath product Sn ࣀ Z2. Using a recently proved bound on the irreducible characters of Sn, we show that no algorithm in this family can solve Graph Isomorphism in less than eΩ(√n) time, no matter what adaptive rule it uses to select and combine states. In particular, algorithms of this type can offer essentially no improvement over the best known classical algorithms, which run in time eO(√(n log n)).