Almost-linear time decoding algorithm for topological codes
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
| Published: | 2021-12-02, volume 5, page 595 |
| Eprint: | arXiv:1709.06218v3 |
| Doi: | https://doi.org/10.22331/q-2021-12-02-595 |
| Citation: | Quantum 5, 595 (2021). |
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Abstract
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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