  We construct a three-dimensional quantum cellular automaton (QCA), an
automorphism of the local operator algebra on a lattice of qubits, which
disentangles the ground state of the Walker-Wang three fermion model. We show
that if this QCA can be realized by a quantum circuit of constant depth, then
there exists a two-dimensional commuting projector Hamiltonian which realizes
the three fermion topological order which is widely believed not to be
possible. We conjecture in accordance with this belief that this QCA is not a
quantum circuit of constant depth, and we provide two further pieces of
evidence to support the conjecture. We show that this QCA maps every local
Pauli operator to a local Pauli operator, but is not a Clifford circuit of
constant depth. Further, we show that if the three-dimensional QCA can be
realized by a quantum circuit of constant depth, then there exists a
two-dimensional QCA acting on fermionic degrees of freedom which cannot be
realized by a quantum circuit of constant depth; i.e., we prove the existence
of a nontrivial QCA in either three or two dimensions. The square of our
three-dimensional QCA can be realized by a quantum circuit of constant depth,
and this suggests the existence of a $\mathbb{Z}_2$ invariant of a QCA in
higher dimensions, totally distinct from the classification by positive
rationals (i.e., by one integer index for each prime) in one dimension.
  In an appendix, unrelated to the main body of this paper, we give a fermionic
generalization of a result of Bravyi and Vyalyi on ground states of 2-local
commuting Hamiltonians.
