Quantum cellular automata — unitary evolutions that preserve locality on a lattice — are fundamental objects in quantum information. They classify topological phases of matter and define the notion of “quantum computation that respects spatial structure.” Separately, coarse homology theories — algebraic invariants of large-scale geometry that ignore small-scale details — are fundamental objects in geometric topology.
These are the same thing (arXiv:2603.10501). Quantum cellular automata naturally constitute the degree-zero component of a coarse homology theory. The Omega-spectrum structure of the QCA space — previously established by Ji and Yang through detailed construction — falls out immediately as a consequence of the axioms of coarse homology.
The structural observation: the classification of quantum cellular automata is not a question about quantum mechanics or computation — it is a question about coarse geometry. The QCA space has the structure it has because it satisfies the axioms of a homology theory, and homology theories are rigid. Once you know the axioms are satisfied, the entire algebraic structure — the spectrum, the suspension isomorphisms, the long exact sequences — follows for free. The detailed construction of Ji and Yang was proving a consequence of a deeper fact. The QCA didn't need to be classified; they needed to be recognized as a homology theory, and the classification follows.