Families of $d=2$ 2D subsystem stabilizer codes for universal Hamiltonian quantum computation with two-body interactions

Phattharaporn Singkanipa; Zihan Xia; Daniel A. Lidar

doi:10.22331/q-2025-08-05-1821

Families of $d=2$ 2D subsystem stabilizer codes for universal Hamiltonian quantum computation with two-body interactions

Phattharaporn Singkanipa^1,2, Zihan Xia^1,3, and Daniel A. Lidar^1,2,3,4

¹Center for Quantum Information Science & Technology
²Department of Physics & Astronomy
³Department of Electrical & Computer Engineering
⁴Department of Chemistry, University of Southern California, Los Angeles, California 90089, USA

Published:	2025-08-05, volume 9, page 1821
Editor:	Alioscia Hamma
Eprint:	arXiv:2412.06744v3
Doi:	https://doi.org/10.22331/q-2025-08-05-1821
Citation:	Quantum 9, 1821 (2025).

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Abstract

In the absence of fault tolerant quantum error correction for analog, Hamiltonian quantum computation, error suppression via energy penalties is an effective alternative. We construct families of distance-$2$ stabilizer subsystem codes we call “trapezoid codes'', that are tailored for energy-penalty schemes. We identify a family of codes achieving the maximum code rate, and by slightly relaxing this constraint, uncover a broader range of codes with enhanced physical locality, thus increasing their practical applicability. Additionally, we provide an algorithm to map the required qubit connectivity graph into graphs compatible with the locality constraints of quantum hardware. Finally, we provide a systematic framework to evaluate the performance of these codes in terms of code rate, physical locality, graph properties, and penalty gap, enabling an informed selection of error-suppression codes for specific quantum computing applications. We identify the $[[4k+2,2k,g,2]]$ family of subsystem codes as optimal in terms of code rate and penalty gap scaling.

Featured image: The induced graphs (required connectivity in quantum devices) to support single-qubit logical operators, two-qubit logical operators, and gauge operators are shown for codes with k = 3 and l = 3.

Popular summary

Hamiltonian quantum computation, including approaches such as adiabatic quantum computing, quantum annealing, and quantum simulation, offers a powerful analog alternative to the standard gate-based quantum computing model. These methods work by continuously changing the energy landscape of a quantum system so that the system naturally evolves toward the solution of a problem. However, because these are analog systems that operate without discrete steps or measurements, incorporating error correction techniques is challenging.

This work introduces new families of quantum error-correcting codes, called trapezoid subsystem codes, that are specifically designed for Hamiltonian-based computation. These codes suppress errors by making them energetically costly, rather than correcting them through active detection. They use only simple two-qubit interactions, making them more practical for real-world devices. Our framework is flexible enough to evaluate each code in the family based on different criteria such as efficiency, locality, and physical layout, enabling researchers to tailor code choices to specific hardware constraints or performance goals.

Our results provide a foundation for building more robust quantum systems without sacrificing hardware feasibility. Future work may explore longer-range error protection, optimize layouts for specific platforms, or adapt the ideas to more complex computational models.

► BibTeX data

@article{Singkanipa2025familiesofdd,
  doi = {10.22331/q-2025-08-05-1821},
  url = {https://doi.org/10.22331/q-2025-08-05-1821},
  title = {Families of {$d=2$} 2{D} subsystem stabilizer codes for universal {H}amiltonian quantum computation with two-body interactions},
  author = {Singkanipa, Phattharaporn and Xia, Zihan and Lidar, Daniel A.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1821},
  month = aug,
  year = {2025}
}

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Cited by

[1] Yingkang Cao, Suying Liu, Haowei Deng, Zihan Xia, Xiaodi Wu, and Yu-Xin Wang, "Robust analog quantum simulators by quantum error-detecting codes", arXiv:2412.07764, (2024).

[2] Victor V. Albert and Philippe Faist, "Handbook of Error-Correcting Codes", arXiv:2606.11484, (2026).

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