Description
A fermionic analogue of a 2D color code.
In the original construction [1], Majorana modes occupy the vertices of a trivalent graph embedded on a cylinder, each face operator is the product of the Majoranas around an even-length face, and the graph is only locally \(3\)-colorable. The code encodes one qubit whose two odd logical operators live on the opposite cylinder boundaries, while an even logical operator is supported on a string connecting the boundaries.
Later hardware-oriented Majorana color-code constructions realize related codes by concatenating Majorana surface codes with small Majorana fermion codes [2–5]. Equivalently, they are multiple Majorana-surface-code layers whose stacked building blocks are replaced by an outer \([[n_f,k,d_m]]_{f}\) Majorana code; hexon, octon, \([[8,3,4]]_{f}\), and \([[10,4,4]]_{f}\) outer codes give concrete order-2, -3, and -4 examples [5].
Protection
For the original cylindrical family with circumference \(R\) and length \(L\), the code distance scales as \(d=\Omega(\min(R,L))\), while the minimum diameter of an even logical operator obeys \(l_{\rm even}=\Omega(L)\) [1; Sec. 7]. The code therefore interpolates between Kitaev-chain-like protection by fermion-parity superselection (\(R=O(1)\), \(L\gg 1\)) and distance-based protection when both linear dimensions are macroscopic.Rate
Concatenating a 4.8.8 Majorana surface code with an outer \([[n_f,k,d_m]]_{f}\) code yields a fermionic mode overhead of \(\frac{2n_f}{k d_m^2} d^2\) per logical qubit of distance \(d\) [5].Gates
Color-to-surface-code lattice surgery [4].Logical tetrons and hexons can be encoded in Majorana color codes and manipulated by ordinary, twist-based, or surface-to-color-code lattice surgery [4,5].Fault Tolerance
Ordinary and twist-based lattice surgery can be made fault tolerant, and surface-to-color-code surgery reduces ancilla-measurement weight for the 16-Majorana-stabilizer families [5] (see also [6]).Cousins
- 2D 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 [1]. Later realizations stack Majorana surface-code layers and replace stacked building blocks with small Majorana fermion codes [2–5].
- Majorana surface 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 [1]. Later realizations stack Majorana surface-code layers and replace stacked building blocks with small Majorana fermion codes [2–5].
- Majorana subsystem stabilizer code— A particular self-dual stabilizer Hamiltonian within the 3D subsystem color code admits a Majorana variant whose boundaries support 2D Majorana color codes [7].
- Majorana box qubit— 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 [5; Sec. V].
- \([[2^{m-1},2^{m-1}-m-1,4]]_{f}\) Hamming Majorana code— The \([[8,3,4]]_{f}\) Hamming Majorana code can replace stacks of three tetrons in a 4.8.8 Majorana surface code to yield an order-3 Majorana color code with maximum stabilizer weight \(16\) [5; Table I].
Primary Hierarchy
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]
- M. B. Hastings, “Small Majorana Fermion Codes”, (2017) arXiv:1703.00612
- [3]
- 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
- [4]
- 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
- [5]
- D. Litinski and F. von Oppen, “Quantum computing with Majorana fermion codes”, Physical Review B 97, (2018) arXiv:1801.08143 DOI
- [6]
- 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
- [7]
- A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
Page edit log
- Victor V. Albert (2024-02-27) — most recent
Cite as:
“Majorana color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/majorana_color