Categorifying Clifford QCA
Bowen Yang
TL;DR
The paper provides a complete classification of Clifford QCAs on arbitrary metric spaces and qudit dimensions by embedding QCA data into the Pedersen–Weibel delooping framework of additive L-theory. The main finding is a natural isomorphism between the stabilization group of Clifford QCAs and the L-theory group $L^1(\mathcal{C}_\Lambda(\mathcal{A}),-1)$, revealing that the classification depends only on the coarse geometry of the underlying space. In Euclidean spaces this recovers known four-fold periodicity phenomena, while for open cones the invariant corresponds to homology with coefficients in the $L$-theory spectrum, linking QCA phases to generalized homology theories. The work further shows how symmetry, mixed qudit dimensions, and coarse-graining fit into the same L-theoretic framework, suggesting broad applications to stable, symmetry-enriched lattice systems and bulk–boundary correspondences.
Abstract
We provide a complete classification of Clifford quantum cellular automata (QCAs) on arbitrary metric spaces and any qudits (of prime or composite dimensions) in terms of algebraic L-theory. Building on the delooping formalism of Pedersen and Weibel, we reinterpret Clifford QCAs as symmetric formations in a filtered additive category constructed from the geometry of the underlying space. This perspective allows us to identify the group of stabilized Clifford QCAs, modulo circuits and separated automorphisms, with the Witt group of the corresponding Pedersen--Weibel category. Notably, because the Pedersen--Weibel category depends only on the large-scale (coarse) structure of the metric space, so too does the classification of Clifford QCAs. For Euclidean lattices, the classification reproduces and expands upon known results, while for more general spaces -- including open cones over finite simplicial complexes -- we relate nontrivial QCAs to generalized homology theories with coefficients in the L-theory spectrum. We also outline extensions to QCAs with symmetry and mixed qudit dimensions, and discuss how these fit naturally into the L-theoretic framework.