On the Impossibility of a Quantum Sieve Algorithm for Graph Isomorphism

TL;DR

This paper provides evidence that quantum sieve algorithms, which have been promising for certain hidden subgroup problems, are unlikely to efficiently solve Graph Isomorphism due to fundamental group-theoretic limitations.

Contribution

It demonstrates that quantum sieve approaches cannot outperform classical algorithms for Graph Isomorphism under a specific group-theoretic conjecture.

Findings

Quantum sieve algorithms cannot solve Graph Isomorphism in subexponential time.

Under a group-theoretic conjecture, quantum approaches are no better than classical algorithms.

The paper links the difficulty of Graph Isomorphism to limitations of quantum measurement strategies.

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 give strong evidence that no such approach can succeed 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. We show, modulo a group-theoretic conjecture regarding the asymptotic characters of the symmetric group, that no matter what rule we use to adaptively combine quantum states, there is a constant b > 0 such that no algorithm in this family can solve Graph Isomorphism in e^{b sqrt{n}} time. In particular, such algorithms are essentially no better than the best known classical algorithms, whose running time is e^{O(sqrt{n \log n})}.