Double-semion string-net code

Double-semion string-net code[1–4]

Description

An \(XS\) stabilizer code that realizes the 2D double semion topological phase. The model can be extended to other spatial dimensions [5].

Cousins

Double-semion stabilizer code— The double-semion stabilizer code and the double-semion string-net code both realize the double semion topological phase, but the former is a modular-qudit stabilizer code, while the latter is an \(XS\) stabilizer code. A commuting-projector version of the double-semion string-net code can also be derived [6,7].
Kitaev surface code— There is a logical basis for both the toric and double-semion string-net codes where each codeword is a superposition of states corresponding to all noncontractible loops of a particular homotopy type. The superposition is equal for the toric code, whereas an odd number of loops appear with a \(-1\) coefficient for the double semion.
Commuting-projector Hamiltonian code— A commuting-projector version of the double-semion string-net code can also be derived [6,7].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

XP stabilizer code

XS stabilizer code

Parents

XS stabilizer code

The double-semion string-net code is an \(XS\) stabilizer code [4; Fig. 1].

Abelian TQD codeAbelian topological Topological Hamiltonian-based QECC Quantum

When treated as ground states of the code Hamiltonian, the double-semion string-net code states realize 2D double-semion topological order, a topological phase of matter that exists as the deconfined phase of the 2D twisted \(\mathbb{Z}_2\) gauge theory [8].

String-net codeTopological Hamiltonian-based QECC Quantum

The string-net model code for the category \(\text{Vec}^{\omega}\mathbb{Z}_2\) for nontrivial cocycle is the double semion string-net code.

Double-semion string-net code

References

[1]: M. A. Levin and X.-G. Wen, “String-net condensation: A physical mechanism for topological phases”, Physical Review B 71, (2005) arXiv:cond-mat/0404617 DOI
[2]: M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
[3]: C. W. von Keyserlingk, F. J. Burnell, and S. H. Simon, “Three-dimensional topological lattice models with surface anyons”, Physical Review B 87, (2013) arXiv:1208.5128 DOI
[4]: X. Ni, O. Buerschaper, and M. Van den Nest, “A non-commuting stabilizer formalism”, Journal of Mathematical Physics 56, (2015) arXiv:1404.5327 DOI
[5]: M. H. Freedman and M. B. Hastings, “Double Semions in Arbitrary Dimension”, Communications in Mathematical Physics 347, 389 (2016) arXiv:1507.05676 DOI
[6]: G. Dauphinais, L. Ortiz, S. Varona, and M. A. Martin-Delgado, “Quantum error correction with the semion code”, New Journal of Physics 21, 053035 (2019) arXiv:1810.08204 DOI
[7]: J. C. Magdalena de la Fuente, N. Tarantino, and J. Eisert, “Non-Pauli topological stabilizer codes from twisted quantum doubles”, Quantum 5, 398 (2021) arXiv:2001.11516 DOI
[8]: R. Dijkgraaf and E. Witten, “Topological gauge theories and group cohomology”, Communications in Mathematical Physics 129, 393 (1990) DOI

Page edit log

Victor V. Albert (2025-07-03) — most recent

Cite as:

“Double-semion string-net code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/double_semion_string_net

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/nonstabilizer/xp_stabilizer/double_semion_string_net copy.yml.