Almost-linear time decoding algorithm for topological codes

Nicolas Delfosse1,2,3 and Naomi H. Nickerson4

1IQIM, California Institute of Technology, Pasadena, CA, USA
2Department of Physics and Astronomy, University of California, Riverside, CA, USA
3Station Q Quantum Architectures and Computation Group, Microsoft Research, Redmond, WA 98052, USA
4Quantum Optics and Laser Science, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ, United Kingdom

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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.

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