CSS code surgery as a universal construction
1Dept. of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK
2Quantinuum, Terrington House, 13-15 Hills Road, Cambridge CB2 1NL, UK
Published: | 2024-05-14, volume 8, page 1344 |
Eprint: | arXiv:2301.13738v6 |
Doi: | https://doi.org/10.22331/q-2024-05-14-1344 |
Citation: | Quantum 8, 1344 (2024). |
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Abstract
We define code maps between Calderbank-Shor-Steane (CSS) codes using maps between chain complexes, and describe code surgery between such codes using a specific colimit in the category of chain complexes. As well as describing a surgery operation, this gives a general recipe for new codes. As an application we describe how to `merge' and `split' along a shared $\overline{X}$ or $\overline{Z}$ operator between arbitrary CSS codes in a fault-tolerant manner, so long as certain technical conditions concerning gauge fixing and code distance are satisfied. We prove that such merges and splits on LDPC codes yield codes which are themselves LDPC.
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Featured image: Diagram representing the universal property of a categorical pushout
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[5] Boldizsár Poór, Razin A. Shaikh, and Quanlong Wang, "ZX-calculus is Complete for Finite-Dimensional Hilbert Spaces", arXiv:2405.10896, (2024).
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