Another subexponential-time quantum algorithm for the dihedral hidden subgroup problem

TL;DR

This paper presents a new subexponential-time quantum algorithm for the dihedral hidden subgroup problem, improving efficiency and flexibility in quantum and classical resource trade-offs over previous algorithms.

Contribution

The paper introduces a more efficient quantum algorithm for the dihedral hidden subgroup problem with adjustable classical and quantum resource trade-offs, extending Regev's algorithm.

Findings

Runs in $ ext{exp}(O( ootrac{ ext{log} N}{})$) quantum time

Uses $ ext{exp}(O( ootrac{ ext{log} N}{})$) classical space with minimal quantum space

Enables multiple hidden shifts and resource trade-offs

Abstract

We give an algorithm for the hidden subgroup problem for the dihedral group $D_N$, or equivalently the cyclic hidden shift problem, that supersedes our first algorithm and is suggested by Regev's algorithm. It runs in $\exp(O(\sqrt{\log N}))$ quantum time and uses $\exp(O(\sqrt{\log N}))$ classical space, but only $O(\log N)$ quantum space. The algorithm also runs faster with quantumly addressable classical space than with fully classical space. In the hidden shift form, which is more natural for this algorithm regardless, it can also make use of multiple hidden shifts. It can also be extended with two parameters that trade classical space and classical time for quantum time. At the extreme space-saving end, the algorithm becomes Regev's algorithm. At the other end, if the algorithm is allowed classical memory with quantum random access, then many trade-offs between classical and quantum time are possible.