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Revisiting finite Abelian hidden subgroup problem and its distributed exact quantum algorithm

Ziyuan Dong, Xiang Fan, Tengxun Zhong, Daowen Qiu

TL;DR

This work advances AHSP by delivering an exact quantum algorithm for any finite Abelian group $G$ when the hidden subgroup order $|K|$ is known, using amplitude amplification to achieve determinism with a complexity of $3( ext{len}(G)- ext{len}(K))$ queries. It then leverages the Chinese Remainder Theorem to construct a distributed exact quantum algorithm (EDK) that reduces per-node qudit requirements and avoids quantum communication by performing parallel local solves and a classical aggregation to recover $K$. A parallel exact classical algorithm (EDCK) is also presented, with improved per-node and total query costs, leveraging Sylow-subgroup decompositions. The results extend to certain non-Abelian groups via direct-product decompositions and illuminate practical paths for NISQ-era distributed quantum computation. Overall, the paper tightly integrates group-theoretic invariants, amplitude amplification, and distributed computing to reduce resource needs while preserving exactness in solving AHSP.

Abstract

We revisit the finite Abelian hidden subgroup problem (AHSP) from a mathematical perspective and make the following contributions. First, by employing amplitude amplification, we present an exact quantum algorithm for the finite AHSP, our algorithm is more concise than the previous exact algorithm and applies to any finite Abelian group. Second, utilizing the Chinese Remainder Theorem, we propose a distributed exact quantum algorithm for finite AHSP, which requires fewer qudits, lower quantum query complexity, and no quantum communication. We further show that our distributed approach can be extended to certain classes of non-Abelian groups. Finally, we develop a parallel exact classical algorithm for finite AHSP with reduced query complexity; even without parallel execution, the total number of queries across all nodes does not exceed that of the original centralized algorithm under mild conditions.

Revisiting finite Abelian hidden subgroup problem and its distributed exact quantum algorithm

TL;DR

This work advances AHSP by delivering an exact quantum algorithm for any finite Abelian group when the hidden subgroup order is known, using amplitude amplification to achieve determinism with a complexity of queries. It then leverages the Chinese Remainder Theorem to construct a distributed exact quantum algorithm (EDK) that reduces per-node qudit requirements and avoids quantum communication by performing parallel local solves and a classical aggregation to recover . A parallel exact classical algorithm (EDCK) is also presented, with improved per-node and total query costs, leveraging Sylow-subgroup decompositions. The results extend to certain non-Abelian groups via direct-product decompositions and illuminate practical paths for NISQ-era distributed quantum computation. Overall, the paper tightly integrates group-theoretic invariants, amplitude amplification, and distributed computing to reduce resource needs while preserving exactness in solving AHSP.

Abstract

We revisit the finite Abelian hidden subgroup problem (AHSP) from a mathematical perspective and make the following contributions. First, by employing amplitude amplification, we present an exact quantum algorithm for the finite AHSP, our algorithm is more concise than the previous exact algorithm and applies to any finite Abelian group. Second, utilizing the Chinese Remainder Theorem, we propose a distributed exact quantum algorithm for finite AHSP, which requires fewer qudits, lower quantum query complexity, and no quantum communication. We further show that our distributed approach can be extended to certain classes of non-Abelian groups. Finally, we develop a parallel exact classical algorithm for finite AHSP with reduced query complexity; even without parallel execution, the total number of queries across all nodes does not exceed that of the original centralized algorithm under mild conditions.
Paper Structure (15 sections, 20 theorems, 85 equations, 5 figures, 3 tables, 9 algorithms)

This paper contains 15 sections, 20 theorems, 85 equations, 5 figures, 3 tables, 9 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Overview of hidden subgroup problem
  4. Properties of $\mathrm{rank}(G)$ and $\mathrm{len}(G)$
  5. Revisiting standard quantum algorithm for finite AHSP
  6. Exact quantum algorithm for finite AHSP
  7. Application of quantum amplitude amplification to finite AHSP
  8. Constructing exact quantum algorithm for finite AHSP
  9. Distributed quantum algorithms for finite AHSP
  10. Subgroup decomposition in direct products and sums
  11. Division of subfunctions
  12. Distributed exact quantum algorithm for finite AHSP
  13. Extensions to non-Abelian groups
  14. Parallel exact classical algorithm for finite AHSP
  15. Conclusion and future works

Key Result

Lemma 1

Let $G$ be a finite Abelian group and $K \leq G$ a subgroup. Then:

Figures (5)

  • Figure 1: The circuit for Algorithm \ref{['algorithm1']}.
  • Figure 2: The circuit for EAHSP (Algorithm \ref{['algorithm3']}).
  • Figure 3: The circuit for DKL in the $i$-th node (Algorithm \ref{['DKL']}).
  • Figure 4: The circuit for EDKL in the $i$-th node (Algorithm \ref{['EDKL']}).
  • Figure 5: The whole circuit for EDK (Algorithm \ref{['EDK']}).

Theorems & Definitions (50)

  • Definition 1: Hidden Subgroup Problem (HSP)kitaev1995quantumnielsen_quantum_2010
  • Remark 1
  • Definition 2: Bilinear Form
  • Definition 3: kaye_introduction_2007
  • Lemma 1: serre1977linearlomont2004hidden
  • Definition 4: hungerford2012algebra
  • Remark 2
  • Proposition 1: dong2025probabilistic
  • Proposition 2: Additivity of Chain Length hungerford2012algebra
  • Proposition 3: Subadditivity of Rank dong2025probabilistic
  • ...and 40 more