Majorana box qubit

Majorana box qubit[1–3]

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].

Gates

Braiding and fusion of MZMs, which act as Ising anyons [5,6].

Decoding

Qubit readout can be done by charge sensing [2,3,6,7].

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].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Fermion code

Majorana stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Parents

Majorana stabilizer code

Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum

Abelian topological codeTopological Hamiltonian-based QECC Quantum

When treated as ground states of the code Hamiltonian, codewords of a single Kitaev chain realize \(\mathbb{Z}_2\) fermionic topological order.

Qubit QLDPC codeStabilizer Hamiltonian-based Qubit QECC Quantum

The Majorana box qubit is a 1D qubit stabilizer code with respect to the Majorana operator basis.

1D lattice stabilizer codeLattice stabilizer Stabilizer Hamiltonian-based QECC Quantum

The Majorana box qubit is a 1D qubit stabilizer code with respect to the Majorana operator basis.

Majorana box qubit

Children

Kitaev chain code

Majorana box qubit codes are obtained from two or more Kitaev-chain code blocks and restricting to the positive-parity logical subspace. The parameter \(n\) in the MBQ code corresponds to the number of Kitaev chains used in the construction.

Tetron code

The Majorana box qubit for \(n=2\) is the tetron code.

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.