Implementing the fanout gate by a Hamiltonian

We show that, for even n, evolving n qubits according to a simple Hamiltonian can be used to exactly implement an (n+1)-qubit parity gate, which is equivalent in constant depth to an (n+1)-qubit fanout gate. We also observe that evolving the Hamiltonian for three qubits results in an inversion-on-three-way-equality gate, which together with single-qubit operations is universal for quantum computation.