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 ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_p$ 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 ${\mathbb{Z}}_2$ Clifford QCAs in $(4l+1)$ spatial dimensions can be disentangled by non-Clifford FDQCs. Our construction applies beyond cubic lattices, allowing ${\mathbb{Z}}_2$ QCAs to be defined on arbitrary cellulations. Furthermore, we explicitly construct invertible subalgebras in higher dimensions, obtaining ${\mathbb{Z}}_2$ ISAs in $2l$ spatial dimensions and ${\mathbb{Z}}_p$ ISAs in $(4l-2)$ spatial dimensions. These ISAs give rise to ${\mathbb{Z}}_2$ QCAs in $(2l+1)$ dimensions and ${\mathbb{Z}}_p$ QCAs in $(4l-1)$ dimensions. We further prove that the QCAs in $3$ spatial dimensions constructed via TQFTs and ISAs are equivalent by identifying their boundary algebras, and show that this approach extends to higher dimensions. Together, these results establish a unified and dimension-periodic framework for Clifford QCAs, connecting their explicit lattice realizations to field theories.