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We present a simple 2d local circuit that implements all-to-all interactionsvia perturbative gadgets. We find an analytic relation between the values$J_{ij}$ of the desired interaction and the parameters of the 2d circuit, aswell as the expression for the error in the quantum spectrum. For the relativeerror 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 andgeneralizes to any hardware. In the architectures available today, $5$ digitsof control precision are sufficient for $n=40,~ \epsilon =0.1$. Comparing ourconstruction, known as paramagnetic trees, to ferromagnetic chains used inminor embedding, we find that at chain length $>3$ the performance of minorembedding degrades exponentially with the length of the chain, while ourconstruction experiences only a polynomial decrease.

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