Clifford quantum cellular automata from topological quantum field theories and invertible subalgebras

We present a general framework for constructing quantum cellular automata (QCA) from topological quantum field theories (TQFT) and invertible subalgebras (ISA) using the cup-product formalism. This approach explicitly realizes all Z₂ and Zₚ Clifford QCAs (for prime p) in all admissible dimensions, in precise agreement with the classification predicted by algebraic L-theory. We determine the orders of these QCAs by explicitly showing that finite powers reduce to the identity up to finite-depth quantum circuits (FDQC) and lattice translations. In particular, we demonstrate that the Z₂ Clifford QCAs in (4l+1) spatial dimensions can be disentangled by non-Clifford FDQCs. Our construction applies beyond cubic lattices, allowing Z₂ QCAs to be defined on arbitrary cellulations. Furthermore, we explicitly construct invertible subalgebras in higher dimensions, obtaining Z₂ ISAs in 2l spatial dimensions and Zₚ ISAs in (4l-2) spatial dimensions. These ISAs give rise to Z₂ QCAs in (2l+1) dimensions and Zₚ QCAs in (4l-1) dimensions. Together, these results establish a unified and dimension-periodic framework for Clifford QCAs, connecting their explicit lattice realizations to field theories.