A Quantum Polynomial-Time Solution to The Dihedral Hidden Subgroup Problem

Matthew Moore; Grace Young

A Quantum Polynomial-Time Solution to The Dihedral Hidden Subgroup Problem

Quantum Physics 2022-02-24 v2 Data Structures and Algorithms

Authors: Matthew Moore , Grace Young

Abstract

We present a polynomial-time quantum algorithm for the Hidden Subgroup Problem over $\mathbb{D}_{2^n}$ . The usual approach to the Hidden Subgroup Problem relies on harmonic analysis in the domain of the problem, and the best known algorithm using this approach has time complexity in $2^{\mathcal{O}(\sqrt{n})}$ . By focusing on structure encoded in the codomain of the problem, we develop a polynomial-time algorithm which uses this structure to direct a "walk" down the subgroup lattice of $\mathbb{D}_{2^n}$ terminating at the hidden subgroup.

Keywords

quantum search algorithms quantum algorithms painlevé equations and orthogonal polynomials

Cite

@article{arxiv.2202.09697,
  title  = {A Quantum Polynomial-Time Solution to The Dihedral Hidden Subgroup Problem},
  author = {Matthew Moore and Grace Young},
  journal= {arXiv preprint arXiv:2202.09697},
  year   = {2022}
}

Comments

Lemma 4.3 is incorrect and cannot be corrected by minor revision. The main algorithm requires the states described in its conclusion, and so must be either reworked or removed

A Quantum Polynomial-Time Solution to The Dihedral Hidden Subgroup Problem

Abstract

Keywords

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