Tight Results on Multiregister Fourier Sampling: Quantum Measurements for Graph Isomorphism Require Entanglement

TL;DR

This paper proves that solving the Graph Isomorphism problem using quantum Fourier sampling requires highly entangled measurements on multiple states, establishing tight lower bounds on the necessary quantum resources.

Contribution

The paper introduces a general method for bounding information extraction via entangled measurements on hidden subgroup states, with tight bounds for the symmetric group case.

Findings

Omega(n log n) lower bound on entangled states needed for non-negligible information

Method applies to the symmetric group relevant for Graph Isomorphism

Results are tight up to a constant factor

Abstract

We establish a general method for proving bounds on the information that can be extracted via arbitrary entangled measurements on tensor products of hidden subgroup coset states. When applied to the symmetric group, the method yields an Omega(n log n) lower bound on the number of coset states over which we must perform an entangled measurement in order to obtain non-negligible information about a hidden involution. These results are tight to within a multiplicative constant and apply, in particular, to the case relevant for the Graph Isomorphism problem. Part of our proof was obtained after learning from Hallgren, Roetteler, and Sen that they had obtained similar results.