### Abstract

We describe a computationally efficient heuristic algorithm based on a renormalization-group procedure which aims at solving the problem of finding a minimal surface given its boundary (curve) in any hypercubic lattice of dimension D > 2. We use this algorithm to correct errors occurring in a four-dimensional variant of the toric code, having open as opposed to periodic boundaries. For a phenomenological error model which includes measurement errors we use a five-dimensional version of our algorithm, achieving a threshold of 4.35±0.1%. For this error model, this is the highest known threshold of any topological code. Without measurement errors, a four-dimensional version of our algorithm can be used and we find a threshold of 7.3±0.1%. For the gate-based depolarizing error model we find a threshold of 0.31±0.01% which is below the threshold found for the twodimensional toric code.

Original language | English |
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Article number | 8528891 |

Pages (from-to) | 2545-2562 |

Number of pages | 18 |

Journal | IEEE Transactions on Information Theory |

Volume | 65 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2019 |

### Keywords

- quantum computing
- error correcting codes
- smoothing method

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## Cite this

*IEEE Transactions on Information Theory*,

*65*(4), 2545-2562. [8528891]. https://doi.org/10.1109/TIT.2018.2879937