Almost-linear time decoding algorithm for topological codes

In order to build a large scale quantum computer, one must be able correct errors extremely fast. We design fast decoding algorithm for topological codes Pauli and erasure combination of both erasure. Our has worst case complexity $O(n \alpha(n))$, where $n$ is the number physical qubits $\alpha$ inverse Ackermann's function, which very slowly growing. For all practical purposes, $\alpha(n) \leq 3$. prove that our performs optimally weight up $(d-1)/2$ loss $d-1$ qubits, $d$ minimum distance code. Numerically, we obtain threshold $9.9\%$ 2d-toric code with perfect syndrome measurements $2.6\%$ faulty measurements.