Towards practical classical processing for the surface code.

The surface code is unarguably the leading quantum error correction code for 2D nearest neighbor architectures, featuring a high threshold error rate of approximately 1%, low overhead implementations of the entire Clifford group, and flexible, arbitrarily long-range logical gates. These highly desirable features come at the cost of significant classical processing complexity. We show how to perform the processing associated with an $n\ifmmode\times\else\texttimes\fi{}n$ lattice of qubits, each being manipulated in a realistic, fault-tolerant manner, in $O({n}^{2})$ average time per round of error correction. We also describe how to parallelize the algorithm to achieve $O(1)$ average processing per round, using only constant computing resources per unit area and local communication. Both of these complexities are optimal.