Here is a list of codes encoding logical information into fermionic systems.

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Code Description
Fermion code Finite-dimensional quantum error-correcting code encoding a logical qudit or fermionic Hilbert space into a physical Fock space of fermionic modes. Codes are typically described using Majorana operators, which are linear combinations of fermionic creation and annihilation operators [1]. Majorana operators may either be considered individually or paired in various ways into creation and annihilation operators to yield fermionic modes. They form a Clifford algebra and can be interpreted as Ising anyons in certain contexts.
Kitaev chain code A Majorana stabilizer code obtained from the ground-state subspace of the Kitaev Majorana chain in its fermionic topological phase [2]. Its codespace is stabilized by nearest-neighbor Majorana bilinears, while two unpaired edge Majoranas furnish one logical fermionic mode. Under parity-preserving noise, it behaves as the Majorana analogue of the repetition code [3].
Majorana box qubit A family of Majorana stabilizer codes obtained by fixing the total fermion parity of \(n\) fermionic modes, equivalently \(2n\) Majorana zero modes, within the ground-state subspace of \(n\) Kitaev Majorana chain Hamiltonians. The resulting positive-parity subspace encodes \(n-1\) logical qubits and has Majorana distance \(2\).
Majorana checkerboard code A Majorana analogue of the X-cube model defined on a cubic lattice. The code admits weight-eight Majorana stabilizer generators on the eight vertices of each cube of a checkerboard sublattice.
Majorana color code A fermionic analogue of a 2D color code.
Majorana stabilizer code A stabilizer code whose stabilizers are products of an even number of Majorana fermion operators, analogous to Pauli strings for a traditional stabilizer code and referred to as Majorana stabilizers. The codespace is the mutual \(+1\) eigenspace of all Majorana stabilizers.
Majorana subsystem stabilizer code Subsystem stabilizer code whose gauge group is generated by products of Majorana operators. Logical information is encoded in physical fermionic modes, with some encoded degrees of freedom designated as gauge qubits.
Majorana surface code Fermionic analogue of the surface code defined on a three-colorable 2D tiling whose face operators are non-overlapping even-Majorana stabilizers. Open patches with four or six alternating colored boundaries encode logical tetrons or hexons. The uniform 4.8.8, 6.6.6, and 4.6.12 tilings yield families with tetron, hexon, or dodecon building blocks and with twist-based lattice surgery supporting minimal-overhead logical Clifford gates [4].
RM Majorana code A Majorana stabilizer code constructed from a self-orthogonal RM code. These codes have the additional property that the global fermion parity is fixed in the codespace. Logical measurements are reduced to parity measurements of some subset of Majorana fermions in the code.
SYK code Approximate \(n\)-fermionic code whose codewords are low-energy states of the Sachdev-Ye-Kitaev (SYK) Hamiltonian [5,6] or other low-rank SYK models [7,8].
Tetron code A \([[2,1,2]]_{f}\) Majorana box qubit encoding a logical qubit into four Majorana modes, equivalently into the fixed-total-parity sector of two physical fermionic modes. Four Majorana zero modes are the smallest aggregate that supports a qubit in a fixed fermion-parity sector [9]. This code can be concatenated with various qubit codes such as surface codes and color codes. Four-boundary Majorana surface-code patches are logical tetrons, i.e., higher-distance analogues of this physical tetron block [4].
Tetron subsystem code Member of a family of Majorana subsystem stabilizer codes on tetrons, obtained from a qubit stabilizer code together with a classical binary code. Tetron parity operators are placed in the gauge group rather than the stabilizer group. This allows the code to correct both odd- and even-weight Majorana errors while using fewer stabilizer generators than the earlier non-subsystem tetron construction with the same capability [10].
\([[2^{m-1},2^{m-1}-m-1,4]]_{f}\) Hamming Majorana code A member of the \([[2^{m-1},2^{m-1}-m-1,4]]_{f}\) family of Majorana stabilizer codes for \(m \geq 3\) constructed from a self-orthogonal first-order RM code (whose dual is the extended Hamming code). A shortened \([[2^{m-1}-1,2^{m-1}-m-2,3]]_{f}\) version can also be defined [11; Prop. 2.5.1]. The logical subspace of the \([[8,3,4]]_{f}\) Hamming Majorana code is a Cartan subspace of the \(E_8\) Lie algebra [12].
\([[6,1,3]]_{f}\) Vijay-Fu Majorana code A Majorana stabilizer code encoding a logical fermion into six physical fermions. This code is the shortest code correcting single fermion-parity flips [13].

List: Union of:

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codes that are descendants of Fermion code

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codes that are descendants of Majorana subsystem stabilizer code

References

[1]
S. B. Bravyi and A. Yu. Kitaev, “Fermionic Quantum Computation”, Annals of Physics 298, 210 (2002) arXiv:quant-ph/0003137 DOI
[2]
A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires”, Physics-Uspekhi 44, 131 (2001) arXiv:cond-mat/0010440 DOI
[3]
A. Schuckert, E. Crane, A. V. Gorshkov, M. Hafezi, and M. J. Gullans, “Fault-tolerant fermionic quantum computing”, (2025) arXiv:2411.08955
[4]
D. Litinski and F. von Oppen, “Quantum computing with Majorana fermion codes”, Physical Review B 97, (2018) arXiv:1801.08143 DOI
[5]
S. Sachdev and J. Ye, “Gapless spin-fluid ground state in a random quantum Heisenberg magnet”, Physical Review Letters 70, 3339 (1993) arXiv:cond-mat/9212030 DOI
[6]
A. Yu. Kitaev, “A simple model of quantum holography (part 2)”, Entanglement in Strongly-Correlated Quantum Matter (2015): 38
[7]
J. Kim, X. Cao, and E. Altman, “Low-rank Sachdev-Ye-Kitaev models”, Physical Review B 101, (2020) arXiv:1910.10173 DOI
[8]
J. Kim, E. Altman, and X. Cao, “Dirac fast scramblers”, Physical Review B 103, (2021) arXiv:2010.10545 DOI
[9]
T. Karzig et al., “Scalable designs for quasiparticle-poisoning-protected topological quantum computation with Majorana zero modes”, Physical Review B 95, (2017) arXiv:1610.05289 DOI
[10]
S. Kundu and B. Reichardt, “Majorana subsystem qubit codes that also correct odd-weight errors”, New Journal of Physics 26, 073029 (2024) DOI
[11]
R. Okada, “A Quantum Analog of Delsarte’s Linear Programming Bounds”, (2025) arXiv:2502.14165
[12]
P. Lévay and F. Holweck, “A fermionic code related to the exceptional group E \({}_{\text{8}}\)”, Journal of Physics A: Mathematical and Theoretical 51, 325301 (2018) arXiv:1801.06998 DOI
[13]
S. Vijay and L. Fu, “Quantum Error Correction for Complex and Majorana Fermion Qubits”, (2017) arXiv:1703.00459