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.

中文翻译：

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拓扑码的近线性时间译码算法

为了建造一台大型量子计算机，必须能够极快地纠正错误。我们为拓扑代码设计了一种快速解码算法，以纠正泡利错误和擦除以及错误和擦除的组合。我们的算法在最坏情况下的复杂度为 $O(n \alpha(n))$，其中 $n$ 是物理量子位的数量，$\alpha$ 是阿克曼函数的逆函数，它的增长非常缓慢。出于所有实际目的，$\alpha(n) \leq 3$。我们证明了我们的算法对于高达 $(d-1)/2$ 的权重错误和高达 $d-1$ 量子比特的损失表现最佳，其中 $d$ 是代码的最小距离。在数值上，我们获得了具有完美综合症测量值的 2d 复曲面代码的阈值 $9.9\%$ 和错误测量值的 $2.6\%$。