Colour Codes Reach Surface Code Performance using Vibe Decoding

Abstract

Two-dimensional quantum colour codes hold significant promise for quantum error correction, offering advantages such as planar connectivity and low overhead logical gates. Despite their theoretical appeal, the practical deployment of these codes faces challenges due to complex decoding requirements compared to surface codes. This paper introduces vibe decoding which, for the first time, brings colour code performance on par with the surface code under practical decoding. Our approach leverages an ensemble of belief propagation decoders - each executing a distinct serial message passing schedule - combined with localised statistics post-processing. We refer to this combined protocol as VibeLSD. The VibeLSD decoder is highly versatile: our numerical results show it outperforms all practical existing colour code decoders across various syndrome extraction schemes, noise models, and error rates. By estimating qubit footprints through quantum memory simulations, we show that colour codes can operate with overhead that is comparable to, and in some cases lower than, that of the surface code. This, combined with the fact that localised statistics decoding is a parallel algorithm, makes VibeLSD suitable for implementation on specialised hardware for real-time decoding. Our results establish the colour code as a practical architecture for near-term quantum hardware, providing improved compilation efficiency for both Clifford and non-Clifford gates without incurring additional qubit overhead relative to the surface code.