A quantum algorithm for the dihedral hidden subgroup problem based on algorithm SV

TL;DR

This paper introduces a quantum algorithm for the dihedral hidden subgroup problem that leverages the shortest vector algorithm, achieving improved space and time complexity over previous methods for certain problem sizes.

Contribution

The paper presents a novel quantum algorithm based on the SV algorithm, reducing space and time complexity for solving the dihedral hidden subgroup problem.

Findings

Uses O(n) quantum space and O(n^2) classical space

Achieves O(n^{0.5} * (log(max a_{ij}))^3) computation time for n<6400

Outperforms existing algorithms with exponential time complexity

Abstract

To accelerate the algorithms for the dihedral hidden subgroup problem, we present a new algorithm based on algorithm SV(shortest vector). A subroutine is given to get a transition quantum state by constructing a phase filter function, then the measurement basis are derived based on the technique for solving low density subset problem. Finally, the parity of slope is revealed by the measurements on the transition quantum state. This algorithm takes O(n) quantum space and O(n^2) classical space, which is superior to existing algorithms, for a relatively small n(n<6400),it takes (n^0.5)*(log(max aij))^3 computation time, which is superior to 2^(O(n^0.5)).