Facilitating practical fault-tolerant quantum computing based on color codes
Open accessAbstract
Color codes are a promising topological code for fault-tolerant quantum computing. Insufficient research on color codes has delayed their practical application. In this work, we address several key issues to facilitate practical fault-tolerant quantum computing based on color codes. First, by introducing decoding graphs with error-rate-related weights, we obtain the threshold of 0.47 % of the 6.6.6 triangular color code under the standard circuit-level noise model, narrowing the gap to that of the surface code. Second, our work first investigates the circuit-level decoding of color code lattice surgery, then gives an efficient decoding algorithm, which is crucial to perform logical operations in a quantum computer with two-dimensional architectures. Last, a state injection protocol of the triangular color code is proposed, reducing the output magic state error rate in one round of 15 to 1 distillation by two orders of magnitude compared to a previous rough protocol. We also prove that our protocol offers the lowest logical error rates for state injection among all possible codes. Published by the American Physical Society 2024
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Colour codes support resource-efficient magic state injection protocols required to implement non-Clifford gates.
“Colour codes can perform all Clifford gates38 within a single error correction cycle and support resource-efficient magic state injection protocols required to implement non-Clifford gates21,39,40.”
From Scaling and logic in the colour code on a superconducting quantum processor (2025) · cites this paper as [39]
Recent advances in superconducting qubit performance, improvements in decoding algorithms and optimized error-syndrome extraction circuits open new possibilities for implementing the colour code on existing hardware.
“Nevertheless, recent advances in superconducting qubit performance8, improvements in decoding algorithms39,40,45–47 and optimized error-syndrome extraction circuits44,47 now open new possibilities for implementing the colour code on existing hardware.”
From Scaling and logic in the colour code on a superconducting quantum processor (2025) · cites this paper as [39]
Lattice surgery enables the efficient realization of multi-qubit fault-tolerant operations, such as the CNOT gate, with only nearest-neighbour interactions.
“Lattice surgery22,39,43,56 enables the efficient realization of multi-qubit fault-tolerant operations, such as the CNOT gate, with only nearest-neighbour interactions.”
From Scaling and logic in the colour code on a superconducting quantum processor (2025) · cites this paper as [39]
Cites
Edmonds, J. (1965) · Canadian Journal of Mathematics · doi:10.4153/cjm-1965-045-4 · Cited by 2,428 · see citing works
Universal quantum computation with ideal Clifford gates and noisy ancillas
Bravyi, S., & Kitaev, A. (2005) · Physical Review A · doi:10.1103/physreva.71.022316 · Cited by 1,525 · cited 1× · see citing works
“Another focus of fault-tolerant quantum computing is the magic state, which is used to perform logical nonClifford gates such as T gates [25].”
MethodQuantum error correction for quantum memories
Terhal, B. M. (2015) · Reviews of Modern Physics · doi:10.1103/revmodphys.87.307 · Cited by 1,354 · see citing works
Surface code quantum computing by lattice surgery
Horsman, D., Fowler, A. G., Devitt, S., & Meter, R. V. (2012) · New Journal of Physics · doi:10.1088/1367-2630/14/12/123011 · Cited by 598 · cited 2× · see citing works
“Lattice surgery [13, 18–21] is a measurement-based scheme allowing for efficient implementation of universal gate sets by fault-tolerant logical multi-body Pauli operator measurements.”
BackgroundFault-tolerant thresholds for quantum error correction with the surface code
Stephens, A. M. (2014) · Physical Review A · doi:10.1103/physreva.89.022321 · Cited by 133 · cited 2× · see citing works
“9% using an efficient decoder [15, 16].”
BackgroundTriangular color codes on trivalent graphs with flag qubits
Chamberland, C., Kubica, A., Yoder, T. J., & Zhu, G. (2020) · New Journal of Physics · doi:10.1088/1367-2630/ab68fd · Cited by 105 · see citing works
A fault-tolerant non-Clifford gate for the surface code in two dimensions
Brown, B. J. (2020) · Science Advances · doi:10.1126/sciadv.aay4929 · Cited by 77 · see citing works
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Scaling and logic in the colour code on a superconducting quantum processor
Lacroix, N., Bourassa, A., Heras, F. J. H., Zhang, L. M., Bausch, J., Senior, A. W., Edlich, T., Shutty, N., Sivak, V., Bengtsson, A., et al. (2025) · Nature · doi:10.1038/s41586-025-09061-4 · Cited by 51 · see citing works
Low-overhead quantum computing with the color code
Thomsen, F., Kesselring, M. S., Bartlett, S. D., & Brown, B. J. (2024) · Physical Review Research · doi:10.1103/physrevresearch.6.043125 · Cited by 15 · see citing works
Improving Threshold for Fault-Tolerant Color-Code Quantum Computing by Flagged Weight Optimization
Takada, Y., & Fujii, K. (2024) · PRX Quantum · doi:10.1103/prxquantum.5.030352 · Cited by 10 · see citing works
- Authors
- Zhang, J., Wu, Y.-C., & Guo, G.-P.
- Affiliations
- University of Science and Technology of China (CN)
- Venue
- Physical Review Research
- Publisher
- American Physical Society (APS)
- Published
- 2024-07-19
- Volume
- 6
- Issue
- 3
- ISSN
- 2643-1564 (ISSN-L)
- OpenAlex
- W4400822994
- Cited by
- 12
- References
- 7
- Citations by year
- 2026: 2 · 2025: 8 · 2024: 2
- FWCI
- 1.78
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Licence
- Locations
- Physical Review Research (publishedVersion, cc-by)
- arXiv (Cornell University) (submittedVersion)
- DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
- arXiv (Cornell University)
- Funders
- National Natural Science Foundation of China (12034018)
- Innovation Program for Quantum Science and Technology (2021ZD0302300)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Keywords
- color codes, fault-tolerant quantum computing, topological quantum error correction, lattice surgery, magic state distillation, logical error rate
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