Quantum Reed-Muller code[1]

Description

A CSS code formed from a classical Reed-Muller code 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]

Parent

Child

Cousins

  • Reed-Muller (RM) code
  • Quantum convolutional code — Quantum convolutional codes can be derived from Quantum Reed-Muller codes [6].
  • Quantum divisible code — Quantum RM codes can be derived using a procedure that yields sufficient conditions for a CSS code to admit a given transversal diagonal logical gate. Quantum divisible codes are derived in a similar procedure, but one that yields necessary and sufficient conditions.
  • Triorthogonal code — Classification of triorthongonal codes yields a connection to Reed-Muller polynomials [7].

Zoo code information

Internal code ID: quantum_reed_muller

Your contribution is welcome!

on github.com (edit & pull request)

edit on this site

Zoo Code ID: quantum_reed_muller

Cite as:
“Quantum Reed-Muller code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_reed_muller
BibTeX:
@incollection{eczoo_quantum_reed_muller, title={Quantum Reed-Muller code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quantum_reed_muller} }
Permanent link:
https://errorcorrectionzoo.org/c/quantum_reed_muller

References

[1]
Andrew Steane, “Quantum Reed-Muller Codes”. quant-ph/9608026
[2]
Shrinivas Kudekar et al., “Reed-Muller Codes Achieve Capacity on Erasure Channels”. 1601.04689
[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). DOI; 1403.2734
[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]
Salah A. Aly, Andreas Klappenecker, and Pradeep Kiran Sarvepalli, “Quantum Convolutional Codes Derived From Reed-Solomon and Reed-Muller Codes”. quant-ph/0701037
[7]
Sepehr Nezami and Jeongwan Haah, “Classification of Small Triorthogonal Codes”. 2107.09684

Cite as:

“Quantum Reed-Muller code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_reed_muller

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