A quantum algorithm for the dihedral hidden subgroup problem based on lattice basis reduction algorithm

Abstract

To optimize the algorithms for the dihedral hidden subgroup problem, we present a new algorithm based on lattice basis reduction algorithm. For n < 120, we reduce the dihedral hidden subgroup problem to shortest vector problem. A subroutine is given to get a transition quantum state by constructing a phase filter function, and then the measurement basis are derived based on the lattice basis reduction algorithm for solving low density subset sum problem. Finally, the parity of slope s is revealed by the measurement. This algorithm needs preparing mn quantum states, m qubits to store and O(n2) classical space, which is superior to existing algorithms.