Almost-linear time decoding algorithm for topological codes

Quantum 5, 595 (2021).

https://doi.org/10.22331/q-2021-12-02-595

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 $O(n alpha(n))$, where $n$ is the number of physical qubits and $alpha$ is the inverse of Ackermann’s function, which is very slowly growing. For all practical purposes, $alpha(n) leq 3$. 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.