Fermionic quantum cellular automata in 2d are trivial

Fermionic quantum cellular automata (QCA) are automorphisms of local fermionic operator algebras (${\mathbb{Z}}_2$-graded superalgebras) that have bounded spread: they map local operators to nearby operators. We prove that every 2-dimensional fermionic QCA on a locally finite-dimensional algebra is a composition of local automorphisms and a fermionic shift. This is implied by our result that every locally finite-dimensional fermionic invertible subalgebra in a one-dimensional lattice is Brauer trivial, $\mathit{\text{i.e.}}$, it is stably bounded-spread isomorphic to a tensor product fermionic algebra.