Designing fault-tolerant circuits using detector error models

Peter-Jan H.S. Derks; Alex Townsend-Teague; Ansgar G. Burchards; Jens Eisert

doi:10.22331/q-2025-11-06-1905

Designing fault-tolerant circuits using detector error models

Peter-Jan H.S. Derks¹, Alex Townsend-Teague¹, Ansgar G. Burchards¹, and Jens Eisert^1,2

¹Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
²Helmholtz-Zentrum Berlin für Materialien und Energie, 14109 Berlin, Germany

Published:	2025-11-06, volume 9, page 1905
Editor:	Carlo Beenakker
Eprint:	arXiv:2407.13826v3
Doi:	https://doi.org/10.22331/q-2025-11-06-1905
Citation:	Quantum 9, 1905 (2025).

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Abstract

Quantum error-correcting codes, such as subspace, subsystem, and Floquet codes, are typically constructed within the stabilizer formalism, which does not fully capture the idea of fault tolerance needed for practical quantum computing applications. In this work, we explore the remarkably powerful formalism of detector error models, which fully captures fault-tolerance at the circuit level. We introduce the detector error model formalism in a pedagogical manner and provide several examples. Additionally, we apply the formalism to three different levels of abstraction in the engineering cycle of fault-tolerant circuit designs: finding robust syndrome extraction circuits, identifying efficient measurement schedules, and constructing fault-tolerant procedures. We enhance the surface code's resistance to measurement errors, devise short measurement schedules for color codes, and implement a more efficient fault-tolerant method for measuring logical operators.

Featured image: Illustration of Clifford circuits at different levels of granularity, whose error-correcting properties can be derived using detector error models. Top left: A syndrome extraction circuit designed for a device with high measurement noise bias. Middle left: A measurement schedule for the [17,1,5] color code. Bottom left: A fault-tolerant protocol for measuring logical operators. Right: Detector error models are composed of a detector error matrix and a noise model.

Popular summary

Imperfect operations present one of the major challenges towards building scalable quantum computers. In general every quantum gate or measurement can introduce small errors to a computation, which over time corrupts the encoded information. While quantum error correction offers a way to protect against such faults, turning theoretical code constructions into resource efficient and practical circuits, executable on specific hardware, remains a major challenge.

This work explores a framework called detector error models, which provides a precise and visual way to understand how faults appear and propagate through quantum circuits. The detector error model framework captures errors at the circuit level, allowing analysis of which combinations of errors can be detected and corrected.

Within this framework, we design surface-code circuits that are more resilient to measurement errors, develop shorter measurement schedules for color codes, and propose an efficient fault-tolerant logical measurement procedure. More broadly, detector error models turn fault-tolerant circuit design into a systematic process, bridging the gap between theoretical codes and quantum hardware.

► BibTeX data

@article{Derks2025designingfault,
  doi = {10.22331/q-2025-11-06-1905},
  url = {https://doi.org/10.22331/q-2025-11-06-1905},
  title = {Designing fault-tolerant circuits using detector error models},
  author = {Derks, Peter-Jan H.S. and Townsend-Teague, Alex and Burchards, Ansgar G. and Eisert, Jens},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1905},
  month = nov,
  year = {2025}
}

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