Here is a list of codes encoding logical information into fermionic systems.
| 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 | An \([[n,1]]_{f}\) 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 Majorana stabilizer code which forms a fixed-parity subspace of the ground-state subspace of one or more Kitaev Majorana chain Hamiltonians. The \([[n,1,2]]_{f}\) Majorana box qubit forms the even-fermion-parity ground-state subspace of two parallel Kitaev Majorana chains in their fermionic topological phase. Its \([[2,1,2]]_{f}\) 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 [4]. |
| 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 or other small Majorana stabilizer 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. |
| 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. |
| \([[2^{m-1},2^{m-1}-m-1,4]]_{f}\) 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 [10; 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 [11]. |
| \([[6,1,3]]_{f}\) 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 [12]. |
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]
- Kitaev, Alexei. “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]
- R. Okada, “A Quantum Analog of Delsarte’s Linear Programming Bounds”, (2025) arXiv:2502.14165
- [11]
- 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
- [12]
- S. Vijay and L. Fu, “Quantum Error Correction for Complex and Majorana Fermion Qubits”, (2017) arXiv:1703.00459