TY - JOUR
T1 - From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups
Y1 - 2005
A1 - Dave Bacon
A1 - Andrew M. Childs
A1 - Wim van Dam
AB - We approach the hidden subgroup problem by performing the so-called pretty good measurement on hidden subgroup states. For various groups that can be expressed as the semidirect product of an abelian group and a cyclic group, we show that the pretty good measurement is optimal and that its probability of success and unitary implementation are closely related to an average-case algebraic problem. By solving this problem, we find efficient quantum algorithms for a number of nonabelian hidden subgroup problems, including some for which no efficient algorithm was previously known: certain metacyclic groups as well as all groups of the form (Z_p)^r X| Z_p for fixed r (including the Heisenberg group, r=2). In particular, our results show that entangled measurements across multiple copies of hidden subgroup states can be useful for efficiently solving the nonabelian HSP.
UR - http://arxiv.org/abs/quant-ph/0504083v2
J1 - Proc. 46th IEEE Symposium on Foundations of Computer Science (FOCS 2005)
U5 - 10.1109/SFCS.2005.38
ER -