# Majorana stabilizer code[1]

## Description

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 \([[n,k,d]]_{f}\) [2], 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 — The Majorana stabilizer code is a stabilizer code whose stabilizers are composed of Majorana fermion operators. In addition, any \([[n,k,d]]\) stabilizer code can be mapped into a \([[2n,k,2d]]_{f}\) Majorana stabilizer code [3][1]. However, Pauli- and Majorana-based stabilizer codes have different notions of locality [4] and are thus useful for different physical platforms.

## Cousins

- Dual linear code — Classical weakly self-dual codes can be used to construct Majorana stabilizer codes [2]. The direct relationship between the two codes follows from expressing the Majorana strings as binary vectors – akin to the binary symplectic representation – and observing that the binary stabilizer matrix \(S\) for such a Majorana stabilizer code satisfies \(S\cdot S^T=0\) because it has commuting stabilizers, which is precisely the condition \(G\cdot G^T=0\) on the generator matrix \(G\) of a weakly self-dual classical code. A weakly self-dual classical code \(C\) with parameters \([2N,k,d]\) yields a Majorana stabilizer code with parameters \([[N,N-k,d^\perp]]_f\), where \(d^\perp\) is the code distance of the dual code \(C^\perp\).
- Calderbank-Shor-Steane (CSS) stabilizer code — When constructing a Majorana stabilizer code from a weakly self-dual classical code with an odd number of bits and generator matrix \(G\), a more complex procedure must be applied to ensure that the fermion code has an even number of Majorana zero modes, and thus a physical Hilbert space [1][2]. Rather than taking \(G\) to be the stabilizer matrix as in the even case, we take \(G\oplus G\). This is a concatenation of classical codes as in the CSS construction and it yields a mapping \([2N-1,k,d]\rightarrow [[2N-1,2N-1-k,d^\perp]]_f\). This procedure may be further generalized by concatenating two different weakly self-dual classical codes with an odd number of bits, as is often done in the CSS construction.
- Cyclic code — Cyclic codes can be used to construct translation-invariant Majorana stabilizer codes, provided that they are also weakly self-dual [2].
- Reed-Muller (RM) code — Majorana stabilizer codes can be constructed by weakly self-dual RM codes [2]. These codes have the additional property that the global fermion parity is fixed in the codespace. In this family of codes, logical measurements are reduced to parity measurements of some subset of Majorana fermions in the code.
- Stabilizer code — Majorana stabilizer codes are useful for Majorana-based architectures, where the degrees of freedom are electrons, and the notion of locality is different than all other code kingdoms.
- Floquet code — Floquet codes are viable candidates for storage in Majorana-qubit devices [5].
- Honeycomb code — The Honeycomb code admits a representation in terms of Majorana fermions. This leads to a possible physical realization of the code in terms of tetrons [6], where each physical qubit is composed of four Majorana modes.
- Transverse-field Ising model (TFIM) code — The TFIM code stabilizers can be expressed in terms of Majorana operators.
- \([[5,1,3]]\) perfect code — \([[5,1,3]]\) code Hamiltonian is local when expressed in terms of mutually commuting Majorana operators [7].

## Zoo code information

## References

- [1]
- S. Bravyi, B. M. Terhal, and B. Leemhuis, “Majorana fermion codes”, New Journal of Physics 12, 083039 (2010). DOI; 1004.3791
- [2]
- Sagar Vijay and Liang Fu, “Quantum Error Correction for Complex and Majorana Fermion Qubits”. 1703.00459
- [3]
- A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006). DOI; cond-mat/0506438
- [4]
- A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires”, Physics-Uspekhi 44, 131 (2001). DOI; cond-mat/0010440
- [5]
- Adam Paetznick et al., “Performance of planar Floquet codes with Majorana-based qubits”. 2202.11829
- [6]
- T. Karzig et al., “Scalable designs for quasiparticle-poisoning-protected topological quantum computation with Majorana zero modes”, Physical Review B 95, (2017). DOI; 1610.05289
- [7]
- Aleksander Kubica, private communication, 2019

## Cite as:

“Majorana stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/majorana_stab

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/fermions/majorana_stab.yml.