# Union-Find Decoders For Homological Product Codes

Nicolas Delfosse1 and Matthew B. Hastings2,1

1Microsoft Quantum and Microsoft Research, Redmond, WA 98052, USA
2Station Q, Microsoft Research, Santa Barbara, CA 93106-6105, USA

### Abstract

Homological product codes are a class of codes that can have improved distance while retaining relatively low stabilizer weight. We show how to build union-find decoders for these codes, using a union-find decoder for one of the codes in the product and a brute force decoder for the other code. We apply this construction to the specific case of the product of a surface code with a small code such as a $[[4,2,2]]$ code, which we call an augmented surface code. The distance of the augmented surface code is the product of the distance of the surface code with that of the small code, and the union-find decoder, with slight modifications, can decode errors up to half the distance. We present numerical simulations, showing that while the threshold of these augmented codes is lower than that of the surface code, the low noise performance is improved.

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### Cited by

[1] Nikolas P. Breuckmann and Jens Niklas Eberhardt, "LDPC Quantum Codes", arXiv:2103.06309.

[2] Nicolas Delfosse, Vivien Londe, and Michael Beverland, "Toward a Union-Find decoder for quantum LDPC codes", arXiv:2103.08049.

[3] Armanda O. Quintavalle and Earl T. Campbell, "Lifting decoders for classical codes to decoders for quantum codes", arXiv:2105.02370.

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