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Classify edge-restricted quantum cellular automata in one dimension

Classify edge-restricted quantum cellular automata in one spatial dimension, i.e., determine all bounded-spread automorphisms of the edge-restricted local operator algebras associated with fusion spin chains arising from matrix-product-operator representations of fusion categorical symmetries.

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Background

The paper studies categorical dualities and their extensions to quantum cellular automata (QCA) using Doplicher–Haag–Roberts (DHR) bimodules. In the categorical symmetry setting, non-unital actions lead naturally to edge-restricted local operator algebras rather than the full quasi-local algebra. Spatial implementations of dualities in this context are edge-restricted QCAs.

The authors note that a full classification of bounded-spread isomorphisms on fusion spin chains can be reduced to classifying edge-restricted QCAs in one spatial dimension, but this classification problem remains unresolved.

References

First, what is the full classification of bounded-spread isomorphisms on local algebras for fusion spin chains? Using our results, we expect to be able to reduce this problem to the classification of edge-restricted QCAs in one spatial dimension, which is an open problem.

— Quantum cellular automata and categorical dualities of spin chains (2410.08884 - Jones et al., 11 Oct 2024) in Introduction, Directions for future work