CSS code surgery as a universal construction

Alexander Cowtan1,2 and Simon Burton2

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

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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.

A well-established method for performing computation with quantum codes based on lattices is lattice surgery, whereby lattices containing quantum data are merged and split apart. In terms of manifolds, we can think of these merges as taking connected sums. Here, we take the natural generalisation of this procedure to Calderbank-Shor-Steane (CSS) codes. The procedure is now based on pushouts, an elementary construction from category theory. We explore the application of such category-theoretic notions to CSS codes, and prove that lattice surgery can be suitably generalised provided certain conditions are satisfied.

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[2] Stephanie Simmons, "Scalable Fault-Tolerant Quantum Technologies with Silicon Color Centers", PRX Quantum 5 1, 010102 (2024).

[3] Boldizsár Poór, Razin A. Shaikh, and Quanlong Wang, "ZX-calculus is Complete for Finite-Dimensional Hilbert Spaces", arXiv:2405.10896, (2024).

[4] Alexander Cowtan, "Towards surgery with good quantum LDPC codes", arXiv:2309.16406, (2023).

[5] Jiaxin Huang, Sarah Meng Li, Lia Yeh, Aleks Kissinger, Michele Mosca, and Michael Vasmer, "Graphical CSS Code Transformation Using ZX Calculus", arXiv:2307.02437, (2023).

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