Description
A family of Majorana stabilizer codes obtained by fixing the total fermion parity of \(n\) fermionic modes, equivalently \(2n\) Majorana zero modes, within the ground-state subspace of \(n\) Kitaev Majorana chain Hamiltonians. The resulting positive-parity subspace encodes \(n-1\) logical qubits and has Majorana distance \(2\).
The \([[2,1,2]]_{f}\) member is called the tetron Majorana code, while 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, the \([[4,3,2]]_{f}\) octon, \([[5,4,2]]_{f}\) decon, and \([[6,5,2]]_{f}\) dodecon are codes defined by the positive-parity subspace of \(4\), \(5\), and \(6\) fermionic modes, respectively [4].
Protection
Fixing total fermion parity detects single-Majorana events as parity violations if that parity is measured. However, these distance-two box-qubit codes do not correct errors: two-Majorana operators act within the fixed-parity sector and can implement logical Pauli errors without changing the stabilizer outcome, while treating the parity constraint as a Hamiltonian term turns single-Majorana events into leakage errors [4].Fault Tolerance
Fault-tolerant computation scheme [8].Cousins
- Hamiltonian-based code— A Majorana box qubit forms a fixed-parity subspace of the ground-state subspace of one or more Kitaev Majorana chain Hamiltonians.
- Majorana surface code— The 4.8.8, 6.6.6, and 4.6.12 Majorana surface-code families realize logical tetrons and hexons as fault-tolerant versions of these small Majorana blocks, using tetrons, hexons, or dodecons as parity-fixed building blocks [4].
- Majorana color code— Majorana color codes are obtained by stacking Majorana surface-code layers and replacing stacked building blocks by small Majorana fermion codes such as hexons, octons, and a \([[10,4,4]]_{f}\) decon-based code [4; Sec. V].
- Kitaev chain code— Majorana box qubit codes are defined to be positive-parity logical subspaces of two or more Kitaev-chain code blocks. The parameter \(n\) in the MBQ code definition corresponds to the number of Kitaev chains used in the construction, and not the total number of physical Majorana modes of the chains.
Primary Hierarchy
References
- [1]
- A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires”, Physics-Uspekhi 44, 131 (2001) arXiv:cond-mat/0010440 DOI
- [2]
- S. Plugge, A. Rasmussen, R. Egger, and K. Flensberg, “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, Y. Oreg, G. Refael, F. von Oppen, and M. P. A. Fisher, “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
- [8]
- D. Aasen et al., “Roadmap to fault tolerant quantum computation using topological qubit arrays”, (2025) arXiv:2502.12252
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/majorana/mbq/mbq.yml.