# Majorana stabilizer code[1]

## Description

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. 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 (equivalently, \(2n\) Majorana modes).

In some cases, Majorana-based stabilizer codes are designed to protect against fermionic noise [3] and are thus useful for physical platforms based on fermions. In other cases, Majorana-based frameworks are helpful for understanding conventional qubit stabilizer codes designed for qubit-based platforms.

## 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.

Code bounds have been developed for small codes [4].

## Encoding

## Gates

## Parents

- Fermion 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 [1,7].

## Children

## 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 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 \(C^\perp\).
- Qubit CSS 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.
- \([2^r-1,2^r-r-1,3]\) Hamming code — Majorana Hamming codes are codes closely related to the classical Hamming codes [4].
- Modular-qudit stabilizer code — Majorana stabilizer codes can be extended to modular qudits, yielding parafermion stabilizer codes [8].
- Hastings-Haah Floquet code — Floquet codes are viable candidates for storage in Majorana-qubit devices [9].
- Honeycomb Floquet code — The Honeycomb code admits a convenient representation in terms of Majorana fermions. This leads to a possible physical realization of the code in terms of tetrons [10], where each physical qubit is composed of four Majorana modes.
- Majorana subsystem stabilizer code — Subsystem qubit stabilizer codes have been formulated in terms of Majorana operators [11].
- Five-qubit perfect code — The five-qubit code Hamiltonian is local when expressed in terms of mutually commuting Majorana operators [12].
- Transverse-field Ising model (TFIM) code — The TFIM code stabilizers can be expressed in terms of Majorana operators.
- Quantum parity code (QPC) — QPCs for \(m_1=m_2\) can be conveniantly expressed in terms of mutually commuting Majorana operators [13].
- Pastawski-Yoshida-Harlow-Preskill (HaPPY) code — The pentagon HaPPY code Hamiltonian can be expressed in terms of mutually commuting weight-two (two-body) Majorana operators [14].
- Kitaev surface code — The Majorana mapping can be used to construct efficient algorithms for simulating rounds of error correction for the surface code [15].

## References

- [1]
- S. Bravyi, B. M. Terhal, and B. Leemhuis, “Majorana fermion codes”, New Journal of Physics 12, 083039 (2010) arXiv:1004.3791 DOI
- [2]
- S. Vijay and L. Fu, “Quantum Error Correction for Complex and Majorana Fermion Qubits”, (2017) arXiv:1703.00459
- [3]
- A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires”, Physics-Uspekhi 44, 131 (2001) arXiv:cond-mat/0010440 DOI
- [4]
- M. B. Hastings, “Small Majorana Fermion Codes”, (2017) arXiv:1703.00612
- [5]
- M. Mudassar, R. W. Chien, and D. Gottesman, “Encoding Majorana codes”, (2024) arXiv:2402.07829
- [6]
- E. Génetay Johansen and T. Simula, “Fibonacci Anyons Versus Majorana Fermions: A Monte Carlo Approach to the Compilation of Braid Circuits in SU(2)k Anyon Models”, PRX Quantum 2, (2021) arXiv:2008.10790 DOI
- [7]
- A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006) arXiv:cond-mat/0506438 DOI
- [8]
- U. Güngördü, R. Nepal, and A. A. Kovalev, “Parafermion stabilizer codes”, Physical Review A 90, (2014) arXiv:1409.4724 DOI
- [9]
- A. Paetznick et al., “Performance of Planar Floquet Codes with Majorana-Based Qubits”, PRX Quantum 4, (2023) arXiv:2202.11829 DOI
- [10]
- 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
- [11]
- A. Chapman, S. T. Flammia, and A. J. Kollár, “Free-Fermion Subsystem Codes”, (2022) arXiv:2201.07254
- [12]
- Aleksander Kubica, private communication, 2019
- [13]
- S. B. Bravyi and A. Yu. Kitaev, “Fermionic Quantum Computation”, Annals of Physics 298, 210 (2002) arXiv:quant-ph/0003137 DOI
- [14]
- A. Jahn et al., “Majorana dimers and holographic quantum error-correcting codes”, Physical Review Research 1, (2019) arXiv:1905.03268 DOI
- [15]
- S. Bravyi et al., “Correcting coherent errors with surface codes”, npj Quantum Information 4, (2018) arXiv:1710.02270 DOI

## Page edit log

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

## Cite as:

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