Description
Quantum-double code whose codewords realize \(G=D_m\) topological order associated with a \(2m\)-element dihedral group \(D_m\). Includes the simplest non-Abelian order \(D_3 = S_3\) associated with the permutation group of three objects. The code can be realized as the ground-state subspace of the quantum double model, defined for \(D_m\)-valued qudits [1]. An alternative qubit-based formulation realizes the gauged \(G=\mathbb{Z}_3^2\) twisted quantum double phase [2], which is the same topological order as the \(G=D_4\) quantum double [3,4].Gates
Universal topological quantum computation is possible for certain groups such as \(G=D_3=S_3\) [5,6].\(U\)-model gate set [7], which can protect from circuit-level noise with the help of an anyon interferometer for the case of \(G=S_3\) [8].Fault Tolerance
Universal topological quantum computation is possible for certain groups such as \(G=D_3=S_3\) [5,6].\(U\)-model gate set [7], which can protect from circuit-level noise with the help of an anyon interferometer for the case of \(G=S_3\) [8].Code Capacity Threshold
Behavior under \(X\)-type noise (namely, diffusion of certain anyons) for the \(G=D_4\) case is related to the phase diagram of a disordered net model [9].Realizations
Signatures of a phase equivalent to the \(G=D_4\) quantum double detected in a 27-qubit trapped-ion device by Quantinuum [10]. Preparation of ground states and braiding of anyons has also been performed. The phase was realized as a gauged \(G=\mathbb{Z}_3^2\) twisted quantum double [2], which is the same topological order as the \(G=D_4\) quantum double [3,4].Notes
See [12][11; Sec. 5.4] for introductions to this code.The \( \Phi, \Lambda \) Decodoku game is based on the quantum double model for the group \(D_3=S_3\) of permutations on three letters.Popular summary of realization of non-Abelian topological order in Quanta Magazine.Cousins
- Abelian TQD stabilizer code— Upon gauging some symmetries [13–15,15], a Type-III \(\mathbb{Z}_2^3\) TQD realizes the same topological order as the \(G=D_4\) quantum double model [2–4].
- Cubic theory code— The ground-state subspace of the cubic theory model code in 2D reduces to that of the gauged \(G=\mathbb{Z}_3^2\) twisted quantum double model [16], realizing the same topological order as the \(G=D_4\) quantum double [3,4].
Primary Hierarchy
Parents
Dihedral \(G=D_m\) quantum-double code
References
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Page edit log
- Victor V. Albert (2023-05-09) — most recent
Cite as:
“Dihedral \(G=D_m\) quantum-double code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/quantum_double_dihedral