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].
Symmetry-protected topological (SPT) code— Gauging [2,8–16] the symmetry of a nontrivial 2D bosonic \(\mathbb{Z}_2\) Ising SPT yields the doubled-semion phase, with semionic \(\pi\)-flux excitations [2; Sec. IV].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Clifford-hierarchy stabilizer code

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 [17].

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]: J. Haegeman, K. Van Acoleyen, N. Schuch, J. I. Cirac, and F. Verstraete, “Gauging Quantum States: From Global to Local Symmetries in Many-Body Systems”, Physical Review X 5, (2015) arXiv:1407.1025 DOI
[9]: S. Vijay, J. Haah, and L. Fu, “Fracton topological order, generalized lattice gauge theory, and duality”, Physical Review B 94, (2016) arXiv:1603.04442 DOI
[10]: D. J. Williamson, “Fractal symmetries: Ungauging the cubic code”, Physical Review B 94, (2016) arXiv:1603.05182 DOI
[11]: L. Bhardwaj, D. Gaiotto, and A. Kapustin, “State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter”, Journal of High Energy Physics 2017, (2017) arXiv:1605.01640 DOI
[12]: A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
[13]: W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
[14]: K. Dolev, V. Calvera, S. S. Cree, and D. J. Williamson, “Gauging the bulk: generalized gauging maps and holographic codes”, Journal of High Energy Physics 2022, (2022) arXiv:2108.11402 DOI
[15]: T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes I. Gauging and dualities”, (2023) arXiv:2310.16032
[16]: D. J. Williamson and T. J. Yoder, “Low-overhead fault-tolerant quantum computation by gauging logical operators”, (2024) arXiv:2410.02213
[17]: 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/clifford_hierarchy/xp/double_semion_string_net.yml.