# 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 self-orthogonal 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 self-orthogonal classical code. A self-orthogonal 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 self-orthogonal 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 self-orthogonal classical codes with an odd number of bits, as is often done in the CSS construction.
- Cyclic linear binary code — Cyclic binary linear codes can be used to construct translation-invariant Majorana stabilizer codes, provided that they are also self-orthogonal [2].
- Reed-Muller (RM) code — Majorana stabilizer codes can be constructed by self-orthogonal 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.
- Five-qubit perfect code — The five-qubit code Hamiltonian is local when expressed in terms of mutually commuting Majorana operators [5].
- Floquet code — Floquet codes are viable candidates for storage in Majorana-qubit devices [6].
- 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 [7], where each physical qubit is composed of four Majorana modes.
- Kitaev surface code — The Majorana mapping can be used to construct efficient algorithms for simulating rounds of error correction for the surface code [8].
- Pastawski-Yoshida-Harlow-Preskill (HaPPY) code — HaPPY code Hamiltonian can be expressed in terms of mutually commuting two-body Majorana operators [9].
- Transverse-field Ising model (TFIM) code — The TFIM code stabilizers can be expressed in terms of Majorana operators.

## 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]
- Aleksander Kubica, private communication, 2019
- [6]
- Adam Paetznick et al., “Performance of planar Floquet codes with Majorana-based qubits”. 2202.11829
- [7]
- 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
- [8]
- S. Bravyi et al., “Correcting coherent errors with surface codes”, npj Quantum Information 4, (2018). DOI; 1710.02270
- [9]
- A. Jahn et al., “Majorana dimers and holographic quantum error-correcting codes”, Physical Review Research 1, (2019). DOI; 1905.03268

## Page edit log

- Victor V. Albert (2022-03-04) — most recent
- Victor V. Albert (2021-12-02)
- Chris Fechisin (2021-11-23)

## Zoo code information

## 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.