How hard is deciding trivial versus nontrivial in the dihedral coset problem?

TL;DR

This paper investigates the complexity of deciding trivial versus nontrivial cases in the dihedral hidden subgroup problem, revealing connections to subset sum problems and quantum sampling, and establishing new hardness results.

Contribution

It introduces a decision version of the dihedral hidden subgroup problem and links its complexity to subset sum and quantum sampling problems, providing new hardness insights.

Findings

Efficient unitaries mapping the basis to the standard basis can solve subset sum with density >1.

Exact decision of membership in the coset subspace implies solving subset sum collision problem.

Implementing the optimal POVM is as hard as quantum sampling subset sum solutions.

Abstract

We study the hardness of the dihedral hidden subgroup problem. It is known that lattice problems reduce to it, and that it reduces to random subset sum with density $> 1$ and also to quantum sampling subset sum solutions. We examine a decision version of the problem where the question asks whether the hidden subgroup is trivial or order two. The decision problem essentially asks if a given vector is in the span of all coset states. We approach this by first computing an explicit basis for the coset space and the perpendicular space. We then look at the consequences of having efficient unitaries that use this basis. We show that if a unitary maps the basis to the standard basis in any way, then that unitary can be used to solve random subset sum with constant density $>1$. We also show that if a unitary can exactly decide membership in the coset subspace, then the collision problem for subset sum can be solved for density $>1$ but approaching $1$ as the problem size increases. This strengthens the previous hardness result that implementing the optimal POVM in a specific way is as hard as quantum sampling subset sum solutions.