Description
A CSS code formed from a classical RM code or its punctured/shortened versions. Such codes often admit transversal logical gates in the Clifford hierarchy.
Ordinary, punctured, or shortened RM codes can be used to construct quantum RM codes. For example, the original construction [1] uses a general RM\((r,m)\) code for the \(X\)-type stabilizers, and an RM\((r-1,m)\) code for the \(Z\)-type stabilizers.
Non-CSS codes can be derived from such codes by modifying the \(X\)-type stabilizers [1].
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 RM codes achieve channel capacity on erasure channels [3].Magic
The family constructed out of shortened RM codes with parameters \([[\sum_{i=w+1}^m \binom{m}{i}, \sum_{i=0}^{w} \binom{m}{i}, \sum_{i=w+1}^{r+1} \binom{r+1}{i}]]\) for integers \(m > 2r\) and \(r > w \geq 0\) yields protocols with an exponent of \(\gamma < 0.678\), with the fewest resource protocol with \(\gamma < 1\) requiring a code with parameters \(\{r,w,m\} = \{19,14,3r+1\}\) such that \(n \approx 2^{58}\) qubits [4; Corr. 1]. This refutes a conjecture that no protocol could achieve \(\gamma < 1\) [5].Transversal Gates
Stabilizer generators are Pauli strings can be defined as acting on subsets of qubits corresponding to subcubes of the Hamming \(n\)-cube (a.k.a. Boolean hypercube) [6]. Transversal \(Z\)-rotations by angles \(\pi/2^k\) acting on subcubes can implement logical multi-controlled-\(Z\) gates [6].The \([[2^m,{m \choose r}, 2^{\min(r,m-r)}]]\) family, where \(r\) divides \(m\), admits diagonal gates in the form of \(Z\)-rotations by angle \(\pi/2^{m/r}\) [8–10][7; Exam. 8 and Thm. 19]. Of these, the sub-family for \(m=2r\) admits logical Clifford group gates via permutations, transversal gates, and fold-transversal gates [11].The family constructed out of shortened RM codes with parameters \([[\sum_{i=w+1}^m \binom{m}{i}, \sum_{i=0}^{w} \binom{m}{i}, \sum_{i=w+1}^{r+1} \binom{r+1}{i}]]\) for integers \(m > 2r\) and \(r > w \geq 0\) admits a transversal gate at the \(\nu\)th level in the hierarchy whenever \(m > \nu r\) [4; Thm. 1].Fault Tolerance
Gate switching protocol for universal computation [12].Fault-tolerant universal computation can be achieved via code switching between the \([[127,1,15]]\) self-dual doubly even punctured quantum RM code and the \([[127,1,7]]\) triply even punctured quantum RM code [11].Cousins
- Reed-Muller (RM) code— Quantum RM codes are constructed from RM codes via the CSS construction. There is a relation between RM code performance against correlated generalizations of multiple-access channels (MACs) and quantum RM code performance against Pauli channels [13].
- Quantum convolutional code— Quantum convolutional codes can be derived from quantum RM codes [14].
- EA qubit stabilizer code— EA versions of quantum RM codes and their quantum tensor-product variants can be constructed [15].
- Quantum tensor-product code— EA versions of quantum RM codes and their quantum tensor-product variants can be constructed [15].
- Quantum divisible code— Fault-tolerant universal computation can be achieved via code switching between the \([[127,1,15]]\) self-dual doubly even punctured quantum RM code and the \([[127,1,7]]\) triply even punctured quantum RM code [11].
- Quantum polar code— There are codes interpolating between quantum RM and quantum polar codes [16].
- Asymmetric quantum code— Asymmetric quantum RM codes have been constructed [17; Lemma 4.1].
- Covariant block quantum code— Quantum RM codes are approximately covariant and nearly saturate certain covariance-performance bounds [18].
- CSS-T code— Certain quantum RM codes are CSS-T codes [19–21].
- Triorthogonal code— Classification of triorthogonal codes yields a connection to RM code polynomials [22].
- Generalized homological-product qubit CSS code— Iterative tensor-product codes can be constructed out of quantum RM codes [23].
Member of code lists
- Asymmetric quantum codes
- Hamiltonian-based codes
- Quantum codes
- Quantum codes based on homological products
- Quantum codes with a rate
- Quantum codes with fault-tolerant gadgets
- Quantum codes with magic-state yield parameters
- Quantum codes with transversal gates
- Quantum CSS codes
- Quantum LDPC codes
- Quantum Reed-Muller codes and friends
- Stabilizer codes
Primary Hierarchy
Generalized homological-product qubit CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum
Parents
Quantum RM codes are special cases of quantum pin codes [24; Sec. II.D].
Prime-qudit RM codes reduce to quantum RM codes when \(q=p=2\).
Quantum Reed-Muller (RM) code
Children
\([[2^D,D,2]]\) hypercube quantum codes are special cases of the \([[2^m,{m \choose r}, 2^r]]\) quantum RM codes for \(m=D\) and \(r=1\) [8–10][7; Exam. 8].
\([[2^r-1,1,3]]\) simplex codes are special cases of the \([[\sum_{i=w+1}^m \binom{m}{i}, \sum_{i=0}^{w} \binom{m}{i}, \sum_{i=w+1}^{r+1} \binom{r+1}{i}]]\) quantum RM codes for \(w=r=0\) and \(m=r-1\) [4; Thm. 1].
\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming codes are quantum RM codes because Hamming and simplex codes are both punctured RM codes.
The \([[2^{2r-1}-1,1,2^r-1]]\) quantum punctured RM codes are special cases of the \([[\sum_{i=w+1}^m \binom{m}{i}, \sum_{i=0}^{w} \binom{m}{i}, \sum_{i=w+1}^{r+1} \binom{r+1}{i}]]\) family for \(m \to 2r-1\), \(w \to 0\), and \(r \to r-1\).
References
- [1]
- A. Steane, “Quantum Reed-Muller Codes”, (1996) arXiv:quant-ph/9608026
- [2]
- L. Zhang and I. Fuss, “Quantum Reed-Muller Codes”, (1997) arXiv:quant-ph/9703045
- [3]
- 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) 1750 (2016) DOI
- [4]
- M. B. Hastings and J. Haah, “Distillation with Sublogarithmic Overhead”, Physical Review Letters 120, (2018) arXiv:1709.03543 DOI
- [5]
- S. Bravyi and J. Haah, “Magic-state distillation with low overhead”, Physical Review A 86, (2012) arXiv:1209.2426 DOI
- [6]
- A. Barg, N. J. Coble, D. Hangleiter, and C. Kang, “Geometric structure and transversal logic of quantum Reed-Muller codes”, (2024) arXiv:2410.07595
- [7]
- N. Rengaswamy, R. Calderbank, M. Newman, and H. D. Pfister, “On Optimality of CSS Codes for Transversal T”, IEEE Journal on Selected Areas in Information Theory 1, 499 (2020) arXiv:1910.09333 DOI
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- E. T. Campbell and M. Howard, “Unified framework for magic state distillation and multiqubit gate synthesis with reduced resource cost”, Physical Review A 95, (2017) arXiv:1606.01904 DOI
- [9]
- E. T. Campbell and M. Howard, “Unifying Gate Synthesis and Magic State Distillation”, Physical Review Letters 118, (2017) arXiv:1606.01906 DOI
- [10]
- J. Haah and M. B. Hastings, “Codes and Protocols for DistillingT, controlled-S, and Toffoli Gates”, Quantum 2, 71 (2018) arXiv:1709.02832 DOI
- [11]
- A. Gong and J. M. Renes, “Computation with quantum Reed-Muller codes and their mapping onto 2D atom arrays”, (2024) arXiv:2410.23263
- [12]
- 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
- [13]
- D. Abdelhadi, C. Sandon, E. Abbe, and R. Urbanke, “Reed-Muller Codes for Quantum Pauli and Multiple Access Channels”, (2025) arXiv:2506.08651
- [14]
- 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
- [15]
- P. J. Nadkarni, P. Jayakumar, A. Behera, and S. S. Garani, “Entanglement-assisted Quantum Reed-Muller Tensor Product Codes”, Quantum 8, 1329 (2024) arXiv:2303.08294 DOI
- [16]
- K. Hidaka, D. Abdelhadi, and R. Urbanke, “Interpolation of Quantum Polar Codes and Quantum Reed-Muller Codes”, (2025) arXiv:2505.22142
- [17]
- P. K. Sarvepalli, A. Klappenecker, and M. Rötteler, “Asymmetric quantum codes: constructions, bounds and performance”, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, 1645 (2009) DOI
- [18]
- Z.-W. Liu and S. Zhou, “Quantum error correction meets continuous symmetries: fundamental trade-offs and case studies”, (2023) arXiv:2111.06360
- [19]
- E. Andrade, J. Bolkema, T. Dexter, H. Eggers, V. Luongo, F. Manganiello, and L. Szramowski, “CSS-T Codes from Reed Muller Codes”, (2025) arXiv:2305.06423
- [20]
- E. Berardini, A. Caminata, and A. Ravagnani, “Structure of CSS and CSS-T quantum codes”, Designs, Codes and Cryptography (2024) arXiv:2310.16504 DOI
- [21]
- E. Camps-Moreno, H. H. López, G. L. Matthews, D. Ruano, R. San-José, and I. Soprunov, “An algebraic characterization of binary CSS-T codes and cyclic CSS-T codes for quantum fault tolerance”, Quantum Information Processing 23, (2024) arXiv:2312.17518 DOI
- [22]
- S. Nezami and J. Haah, “Classification of small triorthogonal codes”, Physical Review A 106, (2022) arXiv:2107.09684 DOI
- [23]
- B. Audoux and A. Couvreur, “On tensor products of CSS Codes”, (2018) arXiv:1512.07081
- [24]
- C. Vuillot and N. P. Breuckmann, “Quantum Pin Codes”, IEEE Transactions on Information Theory 68, 5955 (2022) arXiv:1906.11394 DOI
Page edit log
- Victor V. Albert (2024-03-14) — most recent
- Benjamin Quiring (2021-12-16)
- Victor V. Albert (2021-12-03)
Cite as:
“Quantum Reed-Muller (RM) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/quantum_reed_muller