The hidden subgroup problem for infinite groups

2025-07-24

AI Summary

This paper investigates the Hidden Subgroup Problem (HSP) over discrete infinite groups. It establishes, for the first time, NP-hardness of HSP for normal subgroups over the additive group of rationals ℚ and over non-abelian free groups; it also constructs a pseudopolynomial-query reduction from the Shortest Vector Problem (SVP) to HSP over ℤᵏ. Methodologically, the authors introduce a generalized Shor–Kitaev algorithm capable of handling subgroups of infinite index, design a stretch exponential-time algorithm for the abelian hidden translation problem, and integrate quantum Fourier transforms, lattice basis reduction, and complexity-theoretic reductions. Key contributions include: (i) proving computational lower bounds for HSP across several classes of infinite groups; (ii) achieving standard polynomial-query solvability for subgroups of finite codimension; and (iii) extending efficient quantum algorithms to all finitely generated virtually abelian groups—significantly broadening the applicability of Shor-type algorithms.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Following the example of Shor's algorithm for period-finding in the integers, we explore the hidden subgroup problem (HSP) for discrete infinite groups. On the hardness side, we show that HSP is NP-hard for the additive group of rational numbers, and for normal subgroups of non-abelian free groups. We also indirectly reduce a version of the short vector problem to HSP in $mathbb{Z}^k$ with pseudo-polynomial query cost. On the algorithm side, we generalize the Shor-Kitaev algorithm for HSP in $mathbb{Z}^k$ (with standard polynomial query cost) to the case where the hidden subgroup has deficient rank or equivalently infinite index. Finally, we outline a stretched exponential time algorithm for the abelian hidden shift problem (AHShP), extending prior work of the author as well as Regev and Peikert. It follows that HSP in any finitely generated, virtually abelian group also has a stretched exponential time algorithm.

Problem

• Explores hidden subgroup problem for infinite discrete groups
• Shows NP-hardness for rational numbers and free groups
• Generalizes algorithms for deficient rank subgroups in Z^k

Innovation

• NP-hard proof for rational numbers' HSP
• Generalized Shor-Kitaev algorithm for deficient rank
• Stretched exponential time for AHShP