Three-fermion (3F) subsystem code

Three-fermion (3F) subsystem code[1,2]

Description

2D subsystem stabilizer code whose low-energy excitations realize the three-fermion anyon theory [1,3,4]. One version uses two qubits at each site [2], while other manifestations utilize a single qubit per site and only weight-two (two-body) interactions [1,5]. All are expected to be equivalent to each other via a local constant-depth Clifford circuit.

Cousins

Kitaev surface code— One version of the 3F subsystem code can be obtained from two copies of the square-lattice surface code by gauging out the anyons \(e_1m_1e_2\) and \(e_2m_2\) [2; Sec. 7.4].
Three-fermion (3F) Walker-Wang model code— The (three-dimensional) 3F Walker-Wang model cluster-like state encodes the temporal gate operations on the (two-dimensional) 3F subsystem code into a third spatial dimension [6].
2D color code— The 2D color code is equivalent to two decoupled copies of the 3F code in the sense that the same anyon theory describes the low-energy excitations of both codes [8][7; Appx. B].
XYZ ruby Floquet code— Together, all ISGs of the XYZ ruby Floquet code generate the gauge group of the 3F subsystem code.

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Subsystem qubit codeSubsystem QECC Quantum

Subsystem qubit stabilizer codeSubsystem QECC Quantum

Parents

Subsystem qubit stabilizer code

Lattice subsystem codeSubsystem QECC Quantum

Abelian topological codeTopological Hamiltonian-based QECC Quantum

The 3F code is a 2D subsystem code characterized by 3F topological order [2], which is chiral and modular.

Three-fermion (3F) subsystem code

References

[1]: H. Bombin, M. Kargarian, and M. A. Martin-Delgado, “Interacting anyonic fermions in a two-body color code model”, Physical Review B 80, (2009) arXiv:0811.0911 DOI
[2]: T. D. Ellison, Y.-A. Chen, A. Dua, W. Shirley, N. Tantivasadakarn, and D. J. Williamson, “Pauli topological subsystem codes from Abelian anyon theories”, Quantum 7, 1137 (2023) arXiv:2211.03798 DOI
[3]: E. Rowell, R. Stong, and Z. Wang, “On classification of modular tensor categories”, (2009) arXiv:0712.1377
[4]: H. Bombin, G. Duclos-Cianci, and D. Poulin, “Universal topological phase of two-dimensional stabilizer codes”, New Journal of Physics 14, 073048 (2012) arXiv:1103.4606 DOI
[5]: H. Bombin, “Topological subsystem codes”, Physical Review A 81, (2010) arXiv:0908.4246 DOI
[6]: S. Roberts and D. J. Williamson, “3-Fermion Topological Quantum Computation”, PRX Quantum 5, (2024) arXiv:2011.04693 DOI
[7]: M. S. Kesselring, F. Pastawski, J. Eisert, and B. J. Brown, “The boundaries and twist defects of the color code and their applications to topological quantum computation”, Quantum 2, 101 (2018) arXiv:1806.02820 DOI
[8]: Zhenghan Wang. private communication, 2017.

Page edit log

Nathanan Tantivasadakarn (2024-03-26) — most recent
Victor V. Albert (2021-12-29)

Cite as:

“Three-fermion (3F) subsystem code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/subsystem_three_fermion

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/subsystem/topological/other/subsystem_three_fermion.yml.