[Paper Review] A subexponential-time quantum algorithm for the dihedral hidden subgroup problem

Greg Kuperberg|ArXiv.org|Feb 14, 2003

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper presents a quantum algorithm for the dihedral hidden subgroup problem (DHSP) with subexponential time and query complexity of $2^{O(ar{\sqrt{\log N}})}$, significantly improving upon the classical $O(\sqrt{N})$ query bound. The algorithm uses a quantum character transform followed by iterative qubit pairing to evolve into favorable group representations, enabling direct measurement of the hidden reflection subgroup.

ABSTRACT

We present a quantum algorithm for the dihedral hidden subgroup problem with time and query complexity $O(\exp(C\sqrt{\log N}))$. In this problem an oracle computes a function $f$ on the dihedral group $D_N$ which is invariant under a hidden reflection in $D_N$. By contrast the classical query complexity of DHSP is $O(\sqrt{N})$. The algorithm also applies to the hidden shift problem for an arbitrary finitely generated abelian group. The algorithm begins with the quantum character transform on the group, just as for other hidden subgroup problems. Then it tensors irreducible representations of $D_N$ and extracts summands to obtain target representations. Finally, state tomography on the target representations reveals the hidden subgroup.

Motivation & Objective

• To develop a quantum algorithm that solves the dihedral hidden subgroup problem (DHSP) with better time and query complexity than classical approaches.
• To address the challenge of the DHSP where the hidden subgroup has many conjugates, making standard quantum character transforms ineffective.
• To achieve a favorable trade-off between query complexity and computational time, specifically subexponential rather than exponential.
• To generalize the approach to the hidden shift problem over finitely generated abelian groups.

Proposed method

• The algorithm begins with a quantum character transform, equivalent to the abelian quantum Fourier transform in the case of $D_N$.
• It prepares a quantum state in a representation of $D_N$, which is initially indecipherable but carries information about the hidden subgroup.
• The core technique involves repeatedly pairing two unfavorable qubits to produce a new qubit in a more favorable representation of $D_N$.
• The algorithm targets specific irreducible representations of $D_N$ that allow direct measurement to reveal the hidden reflection.
• The process leverages the structure of the dihedral group and its representations to gradually isolate the hidden subgroup.
• For $N = 2^n$, the algorithm achieves time and query complexity $2^{O(\sqrt{\log N})}$, with a tighter bound of $\widetilde{O}(3^{\sqrt{2\log_3 N}})$ for $N = r^n$.

Experimental results

Research questions

• RQ1 Can a quantum algorithm solve the dihedral hidden subgroup problem with subexponential time complexity?
• RQ2 How can quantum state evolution through representation pairing improve the efficiency of hidden subgroup detection in non-abelian groups?
• RQ3 What is the minimal query complexity achievable for the DHSP, and how does it compare to classical bounds?
• RQ4 Can the algorithm be generalized to other hidden shift problems over finitely generated abelian groups?
• RQ5 What role do specific irreducible representations of $D_N$ play in enabling direct measurement of the hidden subgroup?

Key findings

• The algorithm achieves time and query complexity of $2^{O(\sqrt{10 \log N})}$ for $N = 2^n$, representing a significant improvement over classical $O(\sqrt{N})$ complexity.
• For $N = r^n$ with fixed radix $r$, the complexity is improved to $\widetilde{O}(3^{\sqrt{2\log_3 N}})$, demonstrating a tighter asymptotic bound.
• The algorithm requires $2^{O(\sqrt{\log N})}$ quantum space, which is a major limitation despite the subexponential time.
• The method successfully identifies the hidden reflection subgroup by evolving quantum states into target representations that allow direct measurement.
• The approach generalizes to the hidden shift problem over arbitrary finitely generated abelian groups, extending its applicability.
• The algorithm builds on prior work by Ettinger and Høyer but introduces a novel pairing mechanism to overcome the limitations of the standard character transform in non-abelian cases.