Quantum Reed-Muller code[1] 

Description

A CSS code formed from a classical Reed-Muller code (or its punctured versions) in which polynomials over finite fields encode data. This is done by transforming these polynomials into the stabilizer generator matrices.

Protection

Detects errors on \(d-1\) qubits, corrects errors on \(\left\lfloor (d-1)/2 \right\rfloor\) qubits.

Rate

\(\frac{k}{n}\), where \(k = 2^r - {r \choose t} + 2 \sum_{i=0}^{t-1} {r \choose i}\). Additionally, CSS codes formed from binary Reed-Muller codes achieve channel capacity on erasure channels [2].

Gates

Magic state distillation in all prime dimensions [3]

Fault Tolerance

Gate switching protocol for universal computation [4].

Threshold

Between \(10^{-3}\) and \(10^{-6}\) for depolarizing noise (assuming ideal decoders), see [5]

Parents

Children

Cousins

References

[1]
A. Steane, “Quantum Reed-Muller Codes”, (1996) arXiv:quant-ph/9608026
[2]
S. Kumar, R. Calderbank, and H. D. Pfister, “Reed-muller codes achieve capacity on the quantum erasure channel”, 2016 IEEE International Symposium on Information Theory (ISIT) (2016) DOI
[3]
E. T. Campbell, H. Anwar, and D. E. Browne, “Magic-State Distillation in All Prime Dimensions Using Quantum Reed-Muller Codes”, Physical Review X 2, (2012) DOI
[4]
J. T. Anderson, G. Duclos-Cianci, and D. Poulin, “Fault-Tolerant Conversion between the Steane and Reed-Muller Quantum Codes”, Physical Review Letters 113, (2014) arXiv:1403.2734 DOI
[5]
L. Luo et al., “Fault-tolerance thresholds for code conversion schemes with quantum Reed–Muller codes”, Quantum Science and Technology 5, 045022 (2020) DOI
[6]
S. A. Aly, A. Klappenecker, and P. K. Sarvepalli, “Quantum Convolutional Codes Derived From Reed-Solomon and Reed-Muller Codes”, (2007) arXiv:quant-ph/0701037
[7]
Z.-W. Liu and S. Zhou, “Quantum error correction meets continuous symmetries: fundamental trade-offs and case studies”, (2023) arXiv:2111.06360
[8]
S. Nezami and J. Haah, “Classification of small triorthogonal codes”, Physical Review A 106, (2022) arXiv:2107.09684 DOI
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Zoo Code ID: quantum_reed_muller

Cite as:
“Quantum Reed-Muller code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/quantum_reed_muller
BibTeX:
@incollection{eczoo_quantum_reed_muller, title={Quantum Reed-Muller code}, booktitle={The Error Correction Zoo}, year={2021}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quantum_reed_muller} }
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Cite as:

“Quantum Reed-Muller code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/quantum_reed_muller

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/rm/quantum_reed_muller.yml.