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
- \([[15,1,3]]\) Reed-Muller code — The \([[15,1,3]]\) code is often noted as the 15-qubit quantum Reed-Muller code in the literature.
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
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