Sinkhorn algorithm for quantum permutation groups

TL;DR

This paper presents a Sinkhorn-type algorithm to generate quantum permutation matrices for analyzing graph symmetries and quantum subgroups, aiding in the detection of quantum symmetries in finite graphs.

Contribution

It introduces a novel Sinkhorn algorithm tailored for quantum permutation matrices and applies it to study quantum symmetries of graphs, a new approach in quantum symmetry analysis.

Findings

Algorithm successfully generates quantum permutation matrices.

Method helps determine quantum symmetries of finite graphs.

Provides data and raises questions for future research.

Abstract

We introduce a Sinkhorn-type algorithm for producing quantum permutation matrices encoding symmetries of graphs. Our algorithm generates square matrices whose entries are orthogonal projections onto one-dimensional subspaces satisfying a set of linear relations. We use it for experiments on the representation theory of the quantum permutation group and quantum subgroups of it. We apply it to the question whether a given finite graph (without multiple edges) has quantum symmetries in the sense of Banica. In order to do so, we run our Sinkhorn algorithm and check whether or not the resulting projections commute. We discuss the produced data and some questions for future research arising from it.