Description
Fermionic analogue of the surface code defined on a three-colorable 2D tiling whose face operators are non-overlapping even-Majorana stabilizers. Open patches with four or six alternating colored boundaries encode logical tetrons or hexons. The uniform 4.8.8, 6.6.6, and 4.6.12 tilings yield families with tetron, hexon, or dodecon building blocks and with twist-based lattice surgery supporting minimal-overhead logical Clifford gates [3].Protection
Under quasiparticle-poisoning noise, single Majorana operators flip the syndromes of adjacent stabilizers and can be decoded with surface-code methods. If the shortest logical operator has Majorana weight \(d_m\), then the code corrects up to \(d_m/2-1\) Majorana errors; the corresponding qubit distance is naturally labeled by \(d=d_m/2\) because Pauli errors on tetron/hexon hardware involve pairs of Majoranas [3]. Implementations that treat one color of stabilizers as parity-fixing constraints reduce measured stabilizer weight, at the cost that parity-violating single-Majorana events become leakage unless that color is also measured [3].Rate
For code distance \(d=d_m/2\), the 4.8.8, 6.6.6, and 4.6.12 families require \(4d^2\), \(6d^2+\mathcal{O}(d)\), and \(12d^2+\mathcal{O}(d)\) Majoranas per logical qubit, respectively [3; Table I]. Their maximum stabilizer weights for fault-tolerant lattice surgery can be reduced to \(8\), \(6\), and \(6\) Majoranas, respectively, compared to \(10\) Majoranas for bosonic twist-based surface-code surgery [3].Gates
All logical Clifford gates, including CNOT, can be implemented with zero time overhead by classically tracking them and measuring Pauli products via ordinary and twist-based lattice surgery [3].Surface-code state injection of noisy magic states into logical tetrons [3; Sec. IV.C].Fault Tolerance
Repeated syndrome rounds make ordinary and twist-based lattice surgery fault tolerant against both data and measurement errors [3] (see also [4]).Cousins
- Kitaev surface code— Majorana surface codes map non-uniquely to bosonic surface codes: replacing each tetron in a 4.8.8 code by a qubit yields the square-lattice surface code, while 6.6.6 and 4.6.12 codes map to rotated-square and Kagome-lattice surface-code realizations, respectively [3].
- Majorana color code— The original Majorana color code is a fermionic analogue of a 2D color code in which one Majorana face operator doubles to matching \(X\)- and \(Z\)-type face checks, but the underlying cylinder graph need only be locally \(3\)-colorable and can support odd boundary logical operators [5]. Later realizations stack Majorana surface-code layers and replace stacked building blocks with small Majorana fermion codes [3,6–8].
- Majorana box qubit— 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 [3].
- Tetron code— Four-boundary Majorana surface-code patches are logical tetrons, i.e., higher-distance versions of the tetron code [3].
Primary Hierarchy
Parents
The Majorana surface code is a 2D qubit stabilizer code with respect to the Majorana operator basis.
The Majorana surface code is a 2D qubit stabilizer code with respect to the Majorana operator basis.
Majorana surface code
References
- [1]
- S. Vijay, T. H. Hsieh, and L. Fu, “Majorana Fermion Surface Code for Universal Quantum Computation”, Physical Review X 5, (2015) arXiv:1504.01724 DOI
- [2]
- L. A. Landau, S. Plugge, E. Sela, A. Altland, S. M. Albrecht, and R. Egger, “Towards Realistic Implementations of a Majorana Surface Code”, Physical Review Letters 116, (2016) arXiv:1509.05345 DOI
- [3]
- D. Litinski and F. von Oppen, “Quantum computing with Majorana fermion codes”, Physical Review B 97, (2018) arXiv:1801.08143 DOI
- [4]
- C. McLauchlan and B. Béri, “A new twist on the Majorana surface code: Bosonic and fermionic defects for fault-tolerant quantum computation”, Quantum 8, 1400 (2024) arXiv:2211.11777 DOI
- [5]
- S. Bravyi, B. M. Terhal, and B. Leemhuis, “Majorana fermion codes”, New Journal of Physics 12, 083039 (2010) arXiv:1004.3791 DOI
- [6]
- M. B. Hastings, “Small Majorana Fermion Codes”, (2017) arXiv:1703.00612
- [7]
- D. Litinski, M. S. Kesselring, J. Eisert, and F. von Oppen, “Combining Topological Hardware and Topological Software: Color-Code Quantum Computing with Topological Superconductor Networks”, Physical Review X 7, (2017) arXiv:1704.01589 DOI
- [8]
- D. Litinski and F. von Oppen, “Braiding by Majorana tracking and long-range CNOT gates with color codes”, Physical Review B 96, (2017) arXiv:1708.05012 DOI
Page edit log
- Victor V. Albert (2024-02-27) — most recent
Cite as:
“Majorana surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/majorana_surface