TY - JOUR
T1 - Quantum algorithm for a generalized hidden shift problem
Y1 - 2005
A1 - Andrew M. Childs
A1 - Wim van Dam
AB - Consider the following generalized hidden shift problem: given a function f on {0,...,M-1} x Z_N satisfying f(b,x)=f(b+1,x+s) for b=0,1,...,M-2, find the unknown shift s in Z_N. For M=N, this problem is an instance of the abelian hidden subgroup problem, which can be solved efficiently on a quantum computer, whereas for M=2, it is equivalent to the dihedral hidden subgroup problem, for which no efficient algorithm is known. For any fixed positive epsilon, we give an efficient (i.e., poly(log N)) quantum algorithm for this problem provided M > N^epsilon. The algorithm is based on the "pretty good measurement" and uses H. Lenstra's (classical) algorithm for integer programming as a subroutine.
UR - http://arxiv.org/abs/quant-ph/0507190v1
J1 - Proc. 18th ACM-SIAM Symposium on Discrete Algorithms (SODA 2007)
ER -