Classification of edge-restricted quantum cellular automata in one dimension

Classify edge-restricted quantum cellular automata in one spatial dimension, namely determine all bounded-spread automorphisms of the edge-restricted local operator algebras that arise from non-unital fusion categorical symmetries on one-dimensional spin chains (constructed from a unitary fusion category C, a strongly tensor-generating object X in the dual category, and an indecomposable C-module category M).

Background

The paper develops a categorical criterion for extending bounded-spread isomorphisms (dualities) of symmetric operator algebras to spatial implementations on larger edge-restricted algebras using DHR bimodules. Building on this, the authors outline that a full classification of bounded-spread isomorphisms on fusion spin chains should reduce to a classification of edge-restricted QCAs in one dimension.

Edge-restricted QCAs arise when categorical symmetries are represented by non-unital MPOs; one restricts to the unit MPO sector, producing a non-unital "edge-restricted" algebra. A classification of QCAs on these algebras in 1D remains unknown and is identified explicitly as open.