## Description

An \([[n,1,2]]_{f}\) Majorana stabilizer code forming the even-fermion-parity ground-state subspace of two parallel Kitaev Majorana chains in their fermionic topological phase. The \([[2,1,2]]_{f}\) 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].

## Protection

Single Majorana operators are detectable because they map the even-parity codespace into an odd-parity error space.

## Gates

## Decoding

## Parents

- Majorana stabilizer code
- Small-distance block quantum code
- Abelian topological code — When treated as ground states of the code Hamiltonian, surface codewords realize, codewords of a single Kitaev chain realize \(\mathbb{Z}_2\) fermionic topological order. The MZMs used to define the tetron code act as Ising anyons, which are non-Abelian.

## Child

- Kitaev chain code — Kitaev chain codewords can be obtained by restricting to only one Kitaev chain out of the two chains that define the tetron Majorana code.

## Cousins

- Hamiltonian-based code — The tetron code forms the ground-state subspace of two Kitaev Majorana chain Hamiltonians.
- Majorana color code — Majorana box qubits, such as hexons, tetrons, octons, and decons, are placed onto patches of a 2D lattice to form Majorana color codes [4; Table I].
- Majorana surface code — Majorana box qubits, such as hexons, tetrons, octons, and dodecons, are placed onto patches of a 2D lattice to form Majorana surface codes [4; Table I].

## References

- [1]
- A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires”, Physics-Uspekhi 44, 131 (2001) arXiv:cond-mat/0010440 DOI
- [2]
- S. Plugge et al., “Majorana box qubits”, New Journal of Physics 19, 012001 (2017) arXiv:1609.01697 DOI
- [3]
- 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
- [4]
- D. Litinski and F. von Oppen, “Quantum computing with Majorana fermion codes”, Physical Review B 97, (2018) arXiv:1801.08143 DOI
- [5]
- J. Alicea et al., “Non-Abelian statistics and topological quantum information processing in 1D wire networks”, Nature Physics 7, 412 (2011) arXiv:1006.4395 DOI
- [6]
- D. Aasen et al., “Milestones Toward Majorana-Based Quantum Computing”, Physical Review X 6, (2016) arXiv:1511.05153 DOI
- [7]
- J. F. Steiner and F. von Oppen, “Readout of Majorana qubits”, Physical Review Research 2, (2020) arXiv:2004.02124 DOI

## Page edit log

- Victor V. Albert (2023-03-07) — most recent

## Cite as:

“Majorana box qubit”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/mbq

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/fermions/mbq.yml.