Optimal and efficient decoding of concatenated quantum block codes
Open accessAbstract
We consider the problem of optimally decoding a quantum error correction code---that is, to find the optimal recovery procedure given the outcomes of partial ``check'' measurements on the system. In general, this problem is NP hard. However, we demonstrate that for concatenated block codes, the optimal decoding can be efficiently computed using a message-passing algorithm. We compare the performance of the message-passing algorithm to that of the widespread blockwise hard decoding technique. Our Monte Carlo results using the five-qubit and Steane's code on a depolarizing channel demonstrate significant advantages of the message-passing algorithms in two respects: (i) Optimal decoding increases by as much as 94$%$ the error threshold below which the error correction procedure can be used to reliably send information over a noisy channel; and (ii) for noise levels below these thresholds, the probability of error after optimal decoding is suppressed at a significantly higher rate, leading to a substantial reduction of the error correction overhead.
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The 1D and 2D outer codes have distances two and four respectively, which double and quadruple the inner code distance by utilizing soft information from the inner surface codes.
“These outer codes have distances two and four respectively which, by utilizing the soft information provided by the inner surface codes30, double and quadruple the inner code distance.”
From Yoked surface codes (2025) · cites this paper as [30]
Passing complementary gap information to the outer code helps it identify likely culprit errors.
“Passing this information to the outer code helps it to identify likely culprit errors30.”
From Yoked surface codes (2025) · cites this paper as [30]
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Optical large-scale quantum computation
Fukui, K., & Takeda, S. (2024) · Elsevier eBooks · doi:10.1016/b978-0-323-98378-5.00012-x · Cited by 2 · see citing works
Hierarchical Quantum Error Correction with Hypergraph Product Code and Rotated Surface Code
Haruna, J., & Fujii, K. (2025) · Progress of Theoretical and Experimental Physics · doi:10.1093/ptep/ptaf130
Observation of a fault tolerance threshold with concatenated codes
Sommers, G. M., Foss‐Feig, M., Hayes, D., Huse, D. A., & Gullans, M. J. (2025) · Physical Review Research · doi:10.1103/v477-jw8l
Approximating Degenerate Quantum Maximum Likelihood Decoding Using the Chase Algorithm
Kakizaki, T. (2025) · IEEE International Conference on Quantum Computing and Engineering (QCE) · doi:10.1109/qce65121.2025.10373
Subsystem many-hypercube codes: High-rate concatenated codes with low-weight syndrome measurements
Nakai, R., & Goto, H. (2025) · Physical Review Applied · doi:10.1103/xbzn-vn37
Enhancing decoding performance using efficient error learning
Iyer, P. S., Jain, A., Bartlett, S. D., & Emerson, J. (2026) · Physical Review Applied · doi:10.1103/vtj7-nh6l
- Authors
- Poulin, D.
- Affiliations
- California Institute of Technology (US)
- Corresponding author
- Poulin, David
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 2006-11-22
- Volume
- 74
- Issue
- 5
- ISSN
- 0556-2791 (ISSN-L)
- 1050-2947
- 1094-1622
- OpenAlex
- W2109785828
- Cited by
- 105
- References
- 26
- Citations by year
- 2026: 6 · 2025: 12 · 2024: 7 · 2023: 5 · 2022: 12 · 2021: 8 · 2020: 1 · 2019: 4 · 2018: 9 · 2017: 9 · 2016: 1 · 2015: 4 · 2014: 8 · 2013: 2
- FWCI
- 3.81
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review A (publishedVersion)
- arXiv (Cornell University) (publishedVersion)
- CaltechAUTHORS (California Institute of Technology) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation
- Gordon and Betty Moore Foundation
- California Institute of Technology
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Physical Sciences › Computer Science › Computer Networks and Communications › Error Correcting Code Techniques
- Keywords
- quantum error correction, optimal decoding, message passing algorithm, five-qubit code, Steane code, depolarizing channel, error threshold, decoding overhead, NP-hardness
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