Fast and Accurate Greenberger-Horne-Zeilinger Encoding Using All-to-all Interactions

The $N$-qubit Greenberger-Horne-Zeilinger (GHZ) state is an important resource for quantum technologies. We consider the task of GHZ encoding using all-to-all interactions, which prepares the GHZ state in a special case, and is furthermore useful for quantum error correction, interaction-rate enhancement, and transmitting information using power-law interactions. The naive protocol based on parallelizing CNOT gates takes $\mathrm{O}(1)$-time of Hamiltonian evolution. In this work, we propose a fast protocol that achieves GHZ encoding with high accuracy. The evolution time $\mathrm{O}(\log^2N/N)$ almost saturates the theoretical limit $\Omega(\log N/N)$. Moreover, the final state is close to the ideal encoded one with high fidelity $> 1-10^{-3}$, up to large system sizes $N\lesssim 2000$. The protocol only requires a few stages of time-independent Hamiltonian evolution; the key idea is to use the data qubit as control, and to use fast spin-squeezing dynamics generated by e.g. two-axis-twisting.