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].Gates
Universal topological quantum computation is possible for certain groups such as \(G=D_3=S_3\) [3,4].\(U\)-model gate set [5], which can protect from circuit-level noise with the help of an anyon interferometer for the case of \(G=S_3\) [6].Fault Tolerance
Universal topological quantum computation is possible for certain groups such as \(G=D_3=S_3\) [3,4].\(U\)-model gate set [5], which can protect from circuit-level noise with the help of an anyon interferometer for the case of \(G=S_3\) [6].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 [7].Notes
See [9][8; 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.Cousins
- Abelian TQD code— A Type-III \(\mathbb{Z}_2^3\) TQD realizes the same topological order as the \(G=D_4\) quantum double model [2,10,11].
- Symmetry-protected topological (SPT) code— The \(D_4\) quantum double model can be obtained by gauging [12–21] symmetries of a Type III \(\mathbb{Z}_2^3\) SPT [2,22].
- Hexagonal \(CZ\) code— The ground-state subspace of the hexagonal \(CZ\) code realizes the topological order of the \(G=\mathbb{Z}^3_2\) TQD model [2,23], which is the same topological order as the \(G=D_4\) quantum double [10,11].
- Brickwork \(XS\) stabilizer code— The ground-state subspace of the brickwork \(XS\) stabilizer code realizes the topological order of the \(G=\mathbb{Z}^3_2\) TQD model [2,23], which is the same topological order as the \(G=D_4\) quantum double [10,11].
- Modular-qudit surface code— The \(D_3\) quantum double model can be obtained by gauging [12–21] the charge conjugation symmetry of the \(\mathbb{Z}_3\) surface code [22]. A magic-state preparation routine for the \(\mathbb{Z}_4\) surface code traverses through the \(D_4\) quantum double model [24].
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