Quantum Fanout and GHZ states using spin-exchange interactions

It has been shown that, for any n > 0, if 2n physical qubits (as spin-1/2 particles) are suitably encoded into n logical qubits, then the Heisenberg interactions (either isotropic or nonisotropic) among 2n qubits can be used to exactly implement an (n + 1)-qubit fanout gate using a certain constant depth circuit [arXiv:quant-ph/0407125]. However, the coupling coefficients in the Hamiltonian considered in that paper are assumed to be all equal. In this current work, we generalize these results and show that for all n > 0, 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. More precisely, we show that for any fixed qubit q among the n data qubits and a suitable time T of evolution, the coupling of q with any other qubit should be an odd integer multiple of π/2T. This work resolves a question left open in the paper [arXiv:quant-ph/0407125].