The hidden subgroup problem for infinite groups

TL;DR

This paper investigates the hidden subgroup problem for infinite groups, establishing its computational hardness in some cases and developing algorithms for specific instances, including generalizations of Shor's algorithm.

Contribution

It extends the understanding of HSP to infinite groups, providing complexity results and algorithms for cases with infinite index or deficient rank subgroups.

Findings

HSP is NP-hard for rational numbers and certain non-abelian free groups

Generalized Shor-Kitaev algorithm for infinite index subgroups in b^k

Outlined a stretched exponential time algorithm for abelian hidden shift problem

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.