The Hidden Subgroup Problem for Universal Algebras

• URL: http://arxiv.org/abs/2001.11298v2
• Date: Fri, 1 May 2020 20:14:21 GMT
• Title: The Hidden Subgroup Problem for Universal Algebras
• Authors: Matthew Moore, Taylor Walenczyk
• Abstract summary: The Hidden Subgroup Problem (HSP) is a computational problem which includes as special cases integer factorization, the discrete discrete problem, graph isomorphism, and the shortest vector problem.
• Score: 0.7832189413179361
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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