Precision of quantum simulation of all-to-all coupling in a local architecture

We present a simple 2d local circuit that implements all-to-all interactions via perturbative gadgets. We find an analytic relation between the values $J_{ij}$ of the desired interaction and the parameters of the 2d circuit, as well as the expression for the error in the quantum spectrum. For the relative error to be a constant $\epsilon$, one requires an energy scale growing as $n^6$ in the number of qubits, or equivalently a control precision up to $n^{-6}$. Our proof is based on the Schrieffer-Wolff transformation and generalizes to any hardware. In the architectures available today, $5$ digits of control precision are sufficient for $n=40,~ \epsilon =0.1$. Comparing our construction, known as paramagnetic trees, to ferromagnetic chains used in minor embedding, we find that at chain length $>3$ the performance of minor embedding degrades exponentially with the length of the chain, while our construction experiences only a polynomial decrease.