Nontrivial Quantum Cellular Automata in Higher Dimensions

Understand

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.