Greg Kuperberg

Abstract We consider the hidden subgroup problem (HSP) for infinite groups, beyond the celebrated original cases established by Shor and Kitaev. We prove that HSP is NP-hard in the rational numbers ℚ under addition, as well as for normal subgroups of a non-abelian free group F_k. We can show that HSP in the lattice ℤ^k is uSVP-hard with unary encoding of vectors. On the other hand, HSP in ℤ^k with standard binary encoding of vectors can be solved in BQP, uniformly in the dimension, generalizing the Shor-Kitaev algorithm to infinite-index hidden subgroups. HSP in any fixed, finitely generated, virtually abelian subgroup can also be solved in subexponential time using established quantum algorithms for the abelian hidden shift problem.