Implementing the quantum fanout operation with simple pairwise interactions

It has been shown that, for even $n$, evolving $n$ qubits according to a Hamiltonian that is the sum of pairwise interactions between the particles, can be used to exactly implement an $(n+1)$-qubit fanout gate using a particular constant-depth circuit [arXiv:quant-ph/0309163]. However, the coupling coefficients in the Hamiltonian considered in that paper are assumed to be all equal. In this paper, we generalize these results and show that for all $n$, including odd $n$, one can exactly implement an $(n+1)$-qubit parity gate and hence, equivalently in constant depth an $(n+1)$-qubit fanout gate, using a similar Hamiltonian but with unequal couplings, and we give an exact characterization of which couplings are adequate to implement fanout via the same circuit. We also investigate pairwise couplings that satisfy an inverse square law, giving necessary and sufficient criteria for implementing fanout given spatial arrangements of identical qubits in two and three dimensions subject to this law. We use our criteria to give planar arrangements of four qubits that (together with a target qubit) are adequate to implement $5$-qubit fanout.