Local active error correction from simulated confinement

We refine an old idea for performing fault-tolerant error correction in topological codes by simulating confining interactions between excitations. We implement confinement using an array of local classical processors that measure syndromes, broadcast messages to neighboring processors, and move excitations using received messages. The dynamics of the resulting real-time decoder is geometrically local, homogeneous in spacetime, and self-organized, operating without any form of global control. We prove that below a threshold error rate, it achieves a memory lifetime scaling as a stretched exponential in the linear system size $L$, provided that it has access to $O({\rm polylog}(L))$ noiseless classical bits for each noisy qubit. When applied to the surface code subject to depolarizing noise and measurement errors of equal strength, numerics indicate a threshold at $p_c \approx 1.5\%$.