The Hidden Subgroup Problem for Universal Algebras
Abstract
The Hidden Subgroup Problem (HSP) is a computational problem which includes as special cases integer factorization, the discrete logarithm problem, graph isomorphism, and the shortest vector problem. The celebrated polynomial-time quantum algorithms for factorization and the discrete logarithm are restricted versions of a generic polynomial-time quantum solution to the HSP for abelian groups, but despite focused research no full solution has yet been found. We propose a generalization of the HSP to include arbitrary algebraic structures and analyze this new problem on powers of 2-element algebras. We prove a complete classification of every such power as quantum tractable (i.e. polynomial-time), classically tractable, quantum intractable, and classically intractable. In particular, we identify a class of algebras for which the generalized HSP exhibits super-polynomial speedup on a quantum computer compared to a classical one.
Cite
@article{arxiv.2001.11298,
title = {The Hidden Subgroup Problem for Universal Algebras},
author = {Matthew Moore and Taylor Walenczyk},
journal= {arXiv preprint arXiv:2001.11298},
year = {2020}
}