Abelian TQD stabilizer code[1]
Description
Modular-qudit stabilizer code whose codewords realize a 2D Abelian twisted-quantum-double topological order on composite-dimensional qudits. For every finite Abelian group \(G=\prod_i \mathbb{Z}_{N_i}\) and every product of Type-I and Type-II cocycles, there is a Pauli stabilizer Hamiltonian realizing the corresponding Abelian TQD [1]. Equivalently, these codes exhaust the 2D Abelian topological orders that admit gapped boundaries [1,2].Rate
On a torus, the ground-state Hilbert-space dimension is \(|G|^2\) for underlying group \(G=\prod_i \mathbb{Z}_{N_i}\) [1].Cousin
- Symmetry-protected topological (SPT) code— Gauging [3–12] the \(1\)-form symmetries associated with gauge charges of Abelian TQD stabilizer codes yields Pauli stabilizer models of SPT phases classified by products of Type-I and Type-II cocycles [1].
Primary Hierarchy
Parents
For every finite Abelian group \(G=\prod_i \mathbb{Z}_{N_i}\) and every product of Type-I and Type-II cocycles, there is a 2D modular-qudit Pauli stabilizer Hamiltonian on composite-dimensional qudits realizing the corresponding Abelian TQD [1].
Every Abelian TQD code with Type-I and -II cocycles can be realized as a modular-qudit Pauli stabilizer code by starting from a stack of Abelian quantum double models (it suffices to take \(\prod_i \mathbb{Z}_{N_i^2}\) toric codes) and condensing certain bosonic anyons [1].
Every Abelian TQD code with Type-I and -II cocycles can be realized as a modular-qudit Pauli stabilizer code by starting from a stack of Abelian quantum double models (it suffices to take \(\prod_i \mathbb{Z}_{N_i^2}\) toric codes) and condensing certain bosonic anyons [1].
Abelian TQD stabilizer code
Children
When treated as ground states of the code Hamiltonian, the double-semion stabilizer code states realize 2D double-semion topological order, i.e., the Abelian TQD for \(G=\mathbb{Z}_2\) with nontrivial Type-I cocycle, a topological phase that also exists as the deconfined phase of the 2D twisted \(\mathbb{Z}_2\) gauge theory [1,13].
The anyon theory corresponding to Abelian quantum double codes is defined by an Abelian group and trivial cocycle. Stacks of Abelian quantum double models are the starting point for constructing all Abelian TQD stabilizer codes by condensing bosons; for \(G=\prod_i \mathbb{Z}_{N_i}\), it suffices to use \(\prod_i \mathbb{Z}_{N_i^2}\) quantum doubles [1]. Conversely, every Abelian anyon theory is a subtheory of some Abelian TQD [14; Sec. 6.2]. Upon gauging some symmetries [3–12], Type-I and II \(\mathbb{Z}_2^3\) TQDs realize the same topological order as certain Abelian quantum double models [15,16].
References
- [1]
- T. D. Ellison, Y.-A. Chen, A. Dua, W. Shirley, N. Tantivasadakarn, and D. J. Williamson, “Pauli Stabilizer Models of Twisted Quantum Doubles”, PRX Quantum 3, (2022) arXiv:2112.11394 DOI
- [2]
- J. Kaidi, Z. Komargodski, K. Ohmori, S. Seifnashri, and S.-H. Shao, “Higher central charges and topological boundaries in 2+1-dimensional TQFTs”, SciPost Physics 13, (2022) arXiv:2107.13091 DOI
- [3]
- M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
- [4]
- 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
- [5]
- 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
- [6]
- D. J. Williamson, “Fractal symmetries: Ungauging the cubic code”, Physical Review B 94, (2016) arXiv:1603.05182 DOI
- [7]
- 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
- [8]
- A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
- [9]
- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
- [10]
- 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
- [11]
- T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes I. Gauging and dualities”, (2023) arXiv:2310.16032
- [12]
- D. J. Williamson and T. J. Yoder, “Low-overhead fault-tolerant quantum computation by gauging logical operators”, (2024) arXiv:2410.02213
- [13]
- R. Dijkgraaf and E. Witten, “Topological gauge theories and group cohomology”, Communications in Mathematical Physics 129, 393 (1990) DOI
- [14]
- 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
- [15]
- M. de W. Propitius, “Topological interactions in broken gauge theories”, (1995) arXiv:hep-th/9511195
- [16]
- B. Yoshida, “Topological phases with generalized global symmetries”, Physical Review B 93, (2016) arXiv:1508.03468 DOI
Page edit log
- Victor V. Albert (2023-04-06) — most recent
Cite as:
“Abelian TQD stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/tqd_abelian_stabilizer