The dihedral hidden subgroup problem
Imin Chen, David Sun
TL;DR
This work analyzes the hidden subgroup problem (HSP) for dihedral groups, clarifying why strong Fourier sampling faces obstructions while the standard quantum algorithm can still yield polynomial quantum query complexity in many settings. It details the QFT for finite groups, the structure of irreducible representations of the dihedral group, and how the standard HSP algorithm operates on $D_N$, including why cyclic cases are tractable and why dihedral cases are harder due to flat distributions in 2D irreps. The paper surveys subexponential DHSP algorithms (notably Kuperberg’s sieve), the role of subset-sum reductions, and the optimal measurements (PGM) landscape, culminating in probabilistic no-go results for DCP and a conceptual link to quantum cloning. Collectively, these results illuminate the hardness landscape of DHSP, with implications for lattice-based cryptography, isogeny problems, and quantum information processing, while outlining both limits and avenues for potential quantum algorithms.
Abstract
We give an exposition of the hidden subgroup problem for dihedral groups from the point of view of the standard hidden subgroup quantum algorithm for finite groups. In particular, we recall the obstructions for strong Fourier sampling to succeed, but at the same time, show how the standard algorithm can be modified to establish polynomial quantum query complexity. Finally, we explain a new connection between the dihedral coset problem and cloning of quantum states.