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]
Parent
Children
- \([[2^r-1, 1, 3]]\) quantum Reed-Muller code
- \([[2^r-1, 2^r-2r-1, 3]]\) Hamming-based CSS code — \([[2^r-1, 2^r-2r-1, 3]]\) Hamming-based CSS codes are quantum Reed-Muller codes because they are formed from classical Hamming codes, which are equivalent to RM\((r-2,r)\).
- \([[2^{2r-1}-1,1,2^r-1]]\) quantum punctured Reed-Muller code
Cousins
- Reed-Muller (RM) code
- Quantum convolutional code — Quantum convolutional codes can be derived from Quantum Reed-Muller codes [6].
- Covariant code — Quantum RM codes are approximately covariant and nearly saturate certain covariance-performance bounds [7].
- Triorthogonal code — Classification of triorthongonal codes yields a connection to Reed-Muller polynomials [8].
- 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.
References
- [1]
- A. Steane, “Quantum Reed-Muller Codes”, (1996) arXiv:quant-ph/9608026
- [2]
- S. Kudekar et al., “Reed-Muller Codes Achieve Capacity on Erasure Channels”, (2016) arXiv: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) 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”, (2022) arXiv:2111.06360
- [8]
- S. Nezami and J. Haah, “Classification of small triorthogonal codes”, Physical Review A 106, (2022) arXiv:2107.09684 DOI
Page edit log
- Benjamin Quiring (2021-12-16) — most recent
- Victor V. Albert (2021-12-03)
Cite as:
“Quantum Reed-Muller code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/quantum_reed_muller