Clifford quantum cellular automata: Trivial group in 2D and Witt group in 3D

We study locality preserving automorphisms of operator algebras on D-dimensional uniform lattices prime p-dimensional qudits quantum cellular automata (QCAs), specializing in those that are translation invariant (TI), and map every Pauli matrix to a tensor product matrices (Clifford). associate antihermitian forms the unit determinant over Laurent polynomial rings TI Clifford QCA with lattice boundaries prove form determines up circuits shifts (trivial). It follows 2D is trivial since this case always trivial. Furthermore, we for any D, fourth power present explicit examples nontrivial D = 3 odd p show Witt group finite field Fp subgroup C(D=3,p) all modulo ones. That is, C(D=3,p?1mod4)?Z2×Z2 C(D=3,p?3mod4)?Z4. The found by disentangling ground state commuting Hamiltonian, which constructed coupling layers dimensional toric codes such an exposed surface has anomalous topological order not realizable Hamiltonians strictly two dimensions. In appendix independent main body paper, revisit recent theorem Freedman Hastings two-dimensional QCA, necessarily or invariant, constant depth circuit followed shift. give more direct proof without using ancillas.