Limitations of quantum coset states for graph isomorphism

Sean Hallgren, Cristopher Moore, Martin Rötteler, Alexander Russell, Pranab Sen
2006 Proceedings of the thirty-eighth annual ACM symposium on Theory of computing - STOC '06  
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 [MRS05] that only an
more » ... ly 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 Ω(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 in general it seems hard to implement a given highly entangled measurement. 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.
doi:10.1145/1132516.1132603 dblp:conf/stoc/HallgrenMRRS06 fatcat:wwqoz6j5jzcmhgtrq7uc4ggvka