Here is a list of codes related to quantum Reed-Muller codes.

[Jump to code graph excerpt]

Code Description
Galois-qudit quantum RM code True Galois-qudit stabilizer code constructed from generalized Reed-Muller (GRM) codes via the Galois-qudit Hermitian construction, the Galois-qudit CSS construction, or directly from their parity-check matrices [1][2; Sec. 4.2].
Prime-qudit RM code Modular-qudit stabilizer code constructed from GRM codes or their duals via the modular-qudit CSS construction.
Quantum Reed-Muller (RM) code 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.
\([[15, 7, 3]]\) quantum Hamming code Self-dual quantum Hamming code that admits permutation-based CZ logical gates. The code is constructed using the CSS construction from the \([15,11,3]\) Hamming code and its \([15,4,8]\) dual code.
\([[15,1,3]]\) quantum RM code A \([[15,1,3]]\) quantum Reed-Muller code that is most easily thought of as a tetrahedral 3D color code.
\([[16,6,4]]\) Tesseract color code A (hyperbolic self-dual CSS) 4D color code defined on a tesseract, with stabilizer generators of both types supported on each cube. A \([[16,4,2,4]]\) tesseract subsystem code can be obtained from this code by using two logical qubits as gauge qubits [3].
\([[2^D,D,2]]\) hypercube quantum code Member of a family of codes defined by placing qubits on a \(D\)-dimensional hypercube, \(Z\)-stabilizers on all two-dimensional faces, and an \(X\)-stabilizer on all vertices. These codes realize gates at the \((D-1)\)-st level of the Clifford hierarchy. The measured physical bit string can be post-processed into both a logical output string and stabilizer checks, enabling end-of-circuit error detection directly from classical samples [4]. Higher-distance generalizations include a \([[2^{2D},D,4]]\) hyperoctahedron family and a \([[2^D(2^D+1),D,4]]\) family built from distance-two \(D\)-dimensional toric/surface-code blocks [4]. Various other concatenations give families with increasing distance (see cousins).
\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code Member of a family of self-dual CSS codes constructed from \([2^r-1,2^r-r-1,3]=C_X=C_Z\) Hamming codes and their duals, the simplex codes. The code’s stabilizer generator matrix blocks \(H_{X}\) and \(H_{Z}\) are both the generator matrix for a simplex code. The weight of each stabilizer generator is \(2^{r-1}\).
\([[2^r-1, 2^r-2r-1, 3]]_p\) quantum Hamming code A family of CSS codes extending quantum Hamming codes to prime qudits of dimension \(p\) by expressing the qubit code stabilizers in local-dimension-invariant (LDI) form [5].
\([[2^r-1,1,3]]\) simplex code Member of a color-code family constructed from a punctured first-order RM\((1,m=r)\) code and its even subcode for \(r \geq 3\). Each code transversally implements a diagonal gate at the \((r-1)\)st level of the Clifford hierarchy [6,7]. Each code is a color code defined on a simplex in \(r-1\) dimensions [8,9], where qubits are placed on the vertices, edges, and faces as well as on the simplex itself. The family also admits an XP-stabilizer presentation at precision \(N = 2^{r-2}\) whose generators are symmetric in \(X\) and \(P\), and only \(2r\) such generators are needed to stabilize the codespace [10; Prop. 27].
\([[2^{2r-1}-1,1,2^r-1]]\) quantum punctured RM code A quantum Reed-Muller code constructed from a punctured self-dual RM code and its even subcode for \(r \geq 2\).
\([[4,2,2]]\) Four-qubit code A four-qubit hyperbolic self-dual CSS stabilizer code that is the smallest two-logical-qubit stabilizer code to detect a single-qubit error. It is unique for its parameters [11; Thm. 8][12; ID 9].
\([[7,1,3]]\) Steane code A \([[7,1,3]]\) self-dual CSS code that is the smallest qubit CSS code to correct a single-qubit error [13]. The code is constructed using the classical binary \([7,4,3]\) Hamming code for protecting against both \(X\) and \(Z\) errors.
\([[8,3,2]]\) Smallest interesting color code Smallest 3D color code whose physical qubits lie on vertices of a cube and which admits a (weakly) transversal \(CCZ\) gate. In encoded IQP sampling, the final measurement outcomes determine both the logical sample and stabilizer checks, enabling end-of-circuit error detection or postselected decoding directly from the classical samples [4].

References

[1]
P. K. Sarvepalli and A. Klappenecker, “Nonbinary Quantum Reed-Muller Codes”, (2005) arXiv:quant-ph/0502001
[2]
C.-Y. Lai and C.-C. Lu, “A Construction of Quantum Stabilizer Codes Based on Syndrome Assignment by Classical Parity-Check Matrices”, IEEE Transactions on Information Theory 57, 7163 (2011) arXiv:0712.0103 DOI
[3]
B. W. Reichardt et al., “Demonstration of quantum computation and error correction with a tesseract code”, (2024) arXiv:2409.04628
[4]
D. Hangleiter, M. Kalinowski, D. Bluvstein, M. Cain, N. Maskara, X. Gao, A. Kubica, M. D. Lukin, and M. J. Gullans, “Fault-Tolerant Compiling of Classically Hard Instantaneous Quantum Polynomial Circuits on Hypercubes”, PRX Quantum 6, (2025) arXiv:2404.19005 DOI
[5]
A. J. Moorthy and L. G. Gunderman, “Local-dimension-invariant Calderbank-Shor-Steane Codes with an Improved Distance Promise”, (2021) arXiv:2110.11510
[6]
B. Zeng, H. Chung, A. W. Cross, and I. L. Chuang, “Local unitary versus local Clifford equivalence of stabilizer and graph states”, Physical Review A 75, (2007) arXiv:quant-ph/0611214 DOI
[7]
S. X. Cui, D. Gottesman, and A. Krishna, “Diagonal gates in the Clifford hierarchy”, Physical Review A 95, (2017) arXiv:1608.06596 DOI
[8]
H. Bombin, “Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes”, (2015) arXiv:1311.0879
[9]
B. J. Brown, N. H. Nickerson, and D. E. Browne, “Fault-tolerant error correction with the gauge color code”, Nature Communications 7, (2016) arXiv:1503.08217 DOI
[10]
M. A. Webster, B. J. Brown, and S. D. Bartlett, “The XP Stabiliser Formalism: a Generalisation of the Pauli Stabiliser Formalism with Arbitrary Phases”, Quantum 6, 815 (2022) arXiv:2203.00103 DOI
[11]
E. M. Rains, “Quantum codes of minimum distance two”, (1997) arXiv:quant-ph/9704043
[12]
A. Cross and D. Vandeth, “Small Binary Stabilizer Subsystem Codes”, (2025) arXiv:2501.17447
[13]
B. Shaw, M. M. Wilde, O. Oreshkov, I. Kremsky, and D. A. Lidar, “Encoding one logical qubit into six physical qubits”, Physical Review A 78, (2008) arXiv:0803.1495 DOI