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We show that a quantum architecture with an error correction procedurelimited to geometrically local operations incurs an overhead that grows withthe system size, even if arbitrary error-free classical computation is allowed.In particular, we prove that in order to operate a quantum error correctingcode in 2D at a logical error rate of $\delta$, a space overhead of$\Omega(\sqrt{\log(1/\delta)})$ is needed for any constant depolarizing noise$p > 0$.

A lower bound on the overhead of quantum error correction in low dimensions