Limitations of Quantum Coset States for Graph Isomorphism

TL;DR

This paper demonstrates that using entangled quantum measurements on multiple coset states is necessary to solve graph isomorphism, revealing fundamental limitations of quantum approaches based on coset states.

Contribution

It establishes a lower bound on the number of coset states needed for quantum graph isomorphism algorithms, showing limitations of entangled measurements.

Findings

Entangled measurements on at least a(n  log n) states are required.

Single or pairwise coset states provide exponentially limited information.

Results extend to other groups like GL(n, F_{p^m}) and G^n.

Abstract

It has been known for some time that graph isomorphism reduces to the hidden subgroup problem (HSP). What is more, most exponential speedups in quantum computation are obtained by solving instances of the HSP. A common feature of the resulting algorithms is the use of quantum coset states, which encode the hidden subgroup. An open question has been how hard it is to use these states to solve graph isomorphism. It was recently shown by Moore, Russell, and Schulman that only an exponentially small amount of information is available from one, or a pair of coset states. A potential source of power to exploit are entangled quantum measurements that act jointly on many states at once. We show that entangled quantum measurements on at least \Omega(n log n) coset states are necessary to get useful information for the case of graph isomorphism, matching an information theoretic upper bound. This may be viewed as a negative result because highly entangled measurements seem hard to implement in general. Our main theorem is very general and also rules out using joint measurements on few coset states for some other groups, such as GL(n, F_{p^m}) and G^n where G is finite and satisfies a suitable property.