Welcome to the Fermionic Kingdom.

Fermionic code
Finite-dimensional quantum error-correcting code encoding a logical 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].
Parents:
Finite-dimensional quantum error-correcting code.
Parent of:
Majorana stabilizer code.
Cousins:
Qubit code, Bosonic code.

Majorana stabilizer code[2]
Majorana fermion stabilizer codes are stabilizer codes 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. In such systems, Majorana fermions may either be considered individually or paired into creation and annihilation operators for fermionic modes. Codes can be denoted as \([[n,k,d]]_{f}\) [3], where \(n\) is the number of fermionic modes.
Protection: Detects products of Majorana operators with weight up to \(d-1\). Physically, protects against dephasing errors caused by coupling of fermion density to the environment and bit-flip errors caused by quasiparticle poisoning processes.
Parents:
Fermionic code, Qubit stabilizer code.
Cousins:
Dual linear code, Calderbank-Shor-Steane (CSS) stabilizer code, Cyclic code, Reed-Muller (RM) code, Stabilizer code.
Cousin of:
Floquet code, Honeycomb code, Transverse-field Ising model (TFIM) code, \([[5,1,3]]\) perfect code.

## References

- [1]
- S. B. Bravyi and A. Y. Kitaev, “Fermionic Quantum Computation”, Annals of Physics 298, 210 (2002). DOI; quant-ph/0003137
- [2]
- S. Bravyi, B. M. Terhal, and B. Leemhuis, “Majorana fermion codes”, New Journal of Physics 12, 083039 (2010). DOI; 1004.3791
- [3]
- Sagar Vijay and Liang Fu, “Quantum Error Correction for Complex and Majorana Fermion Qubits”. 1703.00459