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.
Kitaev chain code An \([[n,1,1]]_{f}\) Majorana stabilizer code forming the ground-state of the Kitaev Majorana chain (a.k.a. Kitaev Majorana wire) in its fermionic topological phase, which is unitarily equivalent to the 1D quantum Ising model in the symmetry-breaking phase via the Jordan-Wigner transformation. The code is usually defined using the algebra of two anti-commuting Majorana operators called Majorana zero modes (MZMs) or Majorana edge modes (MEMs). It can be thought of as the Majorana stabilizer analogue of the quantum repetition code, and it encodes a logical fermion because its logical Majorana operator has odd weight [2].
Majorana box qubit An \([[n,1,2]]_{f}\) Majorana stabilizer code forming the even-fermion-parity ground-state subspace of two parallel Kitaev Majorana chains in their fermionic topological phase. The \([[2,1,2]]_{f}\) version is called the tetron Majorana code. An \([[3,2,2]]_{f}\) extension using three Kitaev chains and housing two logical qubits of the same parity is called the hexon Majorana code. Similarly, octon, decon, and dodecon are codes defined by the positive-parity subspace of \(4\), \(5\), and \(6\) fermionic modes, respectively [3].
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 Majorana analogue of the color code defined on a 2D tricolorable lattice and constructed out of Majorana box qubit codes placed on patches of the lattice.
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 surface code Majorana analogue of the surface code defined on a 2D lattice and constructed out of Majorana box qubit codes placed on patches of the lattice.
SYK code Approximate \(n\)-fermionic code whose codewords are low-energy states of the Sachdev-Ye-Kitaev (SYK) Hamiltonian [4,5] or other low-rank SYK models [6,7].
\([[6,1,3]]_{f}\) 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 [8].

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. Schuckert, E. Crane, A. V. Gorshkov, M. Hafezi, and M. J. Gullans, “Fermion-qubit fault-tolerant quantum computing”, (2024) arXiv:2411.08955
[3]
D. Litinski and F. von Oppen, “Quantum computing with Majorana fermion codes”, Physical Review B 97, (2018) arXiv:1801.08143 DOI
[4]
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
[5]
Kitaev, Alexei. "A simple model of quantum holography (part 2)." Entanglement in Strongly-Correlated Quantum Matter (2015): 38.
[6]
J. Kim, X. Cao, and E. Altman, “Low-rank Sachdev-Ye-Kitaev models”, Physical Review B 101, (2020) arXiv:1910.10173 DOI
[7]
J. Kim, E. Altman, and X. Cao, “Dirac fast scramblers”, Physical Review B 103, (2021) arXiv:2010.10545 DOI
[8]
S. Vijay and L. Fu, “Quantum Error Correction for Complex and Majorana Fermion Qubits”, (2017) arXiv:1703.00459