Almost-linear time decoding algorithm for topological codes

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