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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 andoi:10.1145/1132516.1132603 dblp:conf/stoc/HallgrenMRRS06 fatcat:wwqoz6j5jzcmhgtrq7uc4ggvka