The hidden subgroup problem for infinite groups

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.