Classical and Quantum Algorithms for Testing Equivalence of Group Extensions

TL;DR

This paper explores classical and quantum algorithms for testing the equivalence of group extensions, showing classical efficiency for small groups and quantum efficiency for larger black box groups, leveraging the hidden subgroup problem.

Contribution

It introduces a quantum algorithm for equivalence testing of central extensions in black box groups, a problem previously unresolved.

Findings

Classical algorithms are efficient for small groups given by multiplication tables.

Quantum algorithms can efficiently test equivalence in larger black box groups.

The work applies the hidden subgroup problem to group extension equivalence testing.

Abstract

While efficient algorithms are known for solving many important problems related to groups, no efficient algorithm is known for determining whether two arbitrary groups are isomorphic. The particular case of 2-nilpotent groups, a special type of central extension, is widely believed to contain the essential hard cases. However, looking specifically at central extensions, the natural formulation of being "the same" is not isomorphism but rather "equivalence," which requires an isomorphism to preserves the structure of the extension. In this paper, we show that equivalence of central extensions can be computed efficiently on a classical computer when the groups are small enough to be given by their multiplication tables. However, in the model of black box groups, which allows the groups to be much larger, we show that equivalence can be computed efficiently on a quantum computer but not a classical one (under common complexity assumptions). Our quantum algorithm demonstrates a new application of the hidden subgroup problem for general abelian groups.