-The nearest-neighbor gate $e^{i(\pi/4)\sum_{\langle j,k \rangle} Z_j Z_k}$ is implemented by first applying a controlled-$Z$ ($\mathrm{CZ}$) gate between each pair of qubits, followed by local $Z^{-1/2}$ on both qubits, i.e., $\mathrm{CZ}_{jk}Z_j^{-1/2}Z_k^{-1/2}$. All nearest-neighbor two-qubit gates are applied in the following sequence to maintain constant circuit depth as the system size scales up: first, odd horizontal links; then, even horizontal links; followed by odd vertical links and finally even vertical links.
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