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

Abstract

We show that a quantum architecture with an error correction procedure limited to geometrically local operations incurs an overhead that grows with the system size, even if arbitrary error-free classical computation is allowed. In particular, we prove that in order to operate a quantum error correcting code 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 $p>0$.

21 pages