Quantum Lego and XP Stabilizer Codes

Ruohan Shen; Yixu Wang; ChunJun Cao

doi:10.22331/q-2025-10-06-1876

Quantum Lego and XP Stabilizer Codes

Ruohan Shen^1,2, Yixu Wang³, and ChunJun Cao^2,4

¹Department of Physics, Tsinghua University, Beijing 100084, China
²Institute for Quantum Information and Matter California Institute of Technology, 1200 E California Blvd, Pasadena, CA 91125, USA.
³Institute for Advanced Study, Tsinghua University, Beijing 100084, China
⁴Department of Physics, Virginia Tech, Blacksburg, VA, 24061, USA

Published:	2025-10-06, volume 9, page 1876
Editor:	Kishor Bharti
Eprint:	arXiv:2310.19538v2
Doi:	https://doi.org/10.22331/q-2025-10-06-1876
Citation:	Quantum 9, 1876 (2025).

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Abstract

We apply the recent graphical framework of "Quantum Lego" to XP stabilizer codes where the stabilizer group is generally non-Abelian. We show that the idea of operator matching continues to hold for such codes and is sufficient for generating all their XP symmetries provided the resulting code is XP. We provide an efficient classical algorithm for tracking these symmetries under tensor contraction or conjoining. This constitutes a partial extension of the algorithm implied by the Gottesman-Knill theorem beyond Pauli stabilizer states and Clifford operations. Because conjoining transformations generate quantum operations that are universal, the XP symmetries obtained from these algorithms do not uniquely identify the resulting tensors in general. Using this extended framework, we provide examples of novel XP stabilizer codes with a higher distance than existing non-trivial XP regular codes and a $[[8,1,2]]$ Pauli stabilizer code with a fault-tolerant $T$ gate. For XP regular codes, we also construct a tensor-network-based maximum likelihood decoder for any independently and identically distributed single qubit error channel using weight enumerators.

Popular summary

A new way to generate quantum codes with non-Abelian symmetries called XP stabilizer codes has been identified in this work where small lego blocks of codes are glued together to build up bigger codes. We identify new instances of XP codes using this technique and develop efficient optimal decoders for XP codes built from the smaller "quantum lego blocks".

► BibTeX data

@article{Shen2025quantumlegoxp,
  doi = {10.22331/q-2025-10-06-1876},
  url = {https://doi.org/10.22331/q-2025-10-06-1876},
  title = {Quantum {L}ego and {XP} {S}tabilizer {C}odes},
  author = {Shen, Ruohan and Wang, Yixu and Cao, ChunJun},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1876},
  month = oct,
  year = {2025}
}

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