Alternative names: \(\mathbb{Z}_q\) surface code.
Description
Extension of the surface code to prime-dimensional [1,2] and more general modular qudits [3]. Stabilizer generators are few-body \(X\)-type and \(Z\)-type Pauli strings associated to the stars and plaquettes, respectively, of a tessellation of a two-dimensional surface. Since qudits have more than one \(X\) and \(Z\)-type operator, various sets of stabilizer generators can be defined. Ground-state degeneracy and the associated phase depends on the qudit dimension and the stabilizer generators.Gates
A magic-state preparation routine for the \(\mathbb{Z}_4\) surface code traverses through the \(D_4\) quantum double model [4].Realizations
State preparation, anyon creation, anyon fusion, and transfer of entanglement between anyons and defect in a 24 qubit trapped ion device by Quantinuum [7].Notes
The simplest Decodoku game is based on the qudit surface code with \( q=10\). See related Qiskit tutorial.Cousins
- Dihedral \(G=D_m\) quantum-double code— The \(D_3\) quantum double model can be obtained by gauging [8–17] the charge conjugation symmetry of the \(\mathbb{Z}_3\) surface code [18]. A magic-state preparation routine for the \(\mathbb{Z}_4\) surface code traverses through the \(D_4\) quantum double model [4].
- Hopf-algebra quantum-double code— The modular-qudit surface code can be generalized to a Hopf-algebra quantum-double code whose ground states remain the same but whose excitations are based on quasitriangular semisimple Hopf algebras of \(\mathbb{Z}_q\) [19].
- Compactified \(\mathbb{R}\) gauge theory code— The tiger surface code can be thought of as a realization of the \(q\to\infty\) \(U(1)\) rotor limit [20] of the qudit surface code as a bosonic stabilizer code.
- Analog surface code— The tiger surface code can be thought of as a realization of the \(q\to\infty\) \(\mathbb{R}\) oscillator limit [20] of the qudit surface code as a bosonic stabilizer code.
- Tiger surface code— The tiger surface code can be thought of as a realization of the \(q\to\infty\) \(U(1)\) rotor limit [20] of the qudit surface code as a tiger code.
- Twist-defect surface code— Twist-defect surface codes have been extended to prime-dimensional qudits [21].
- Qudit X-cube model code— A field-theoretic description of the qudit X-cube model can be obtained by coupling layers of 2D \(\mathbb{Z}_q\) gauge theory [22].
- Double-semion stabilizer code— The exchange statistics of the anyon for the double-semion code coincides with a subset of anyons in the \(\mathbb{Z}_4\) surface code, but the fusion rules are different. The double-semion code can be obtained from the \(\mathbb{Z}_4\) surface code by condensing the anyon \(e^2 m^2\) [23] or by gauging [8–17] the one-form symmetry associated with said anyon [23; Footnote 20].
- \(\mathbb{Z}_q^{(1)}\) subsystem code— The \(\mathbb{Z}_q^{(1)}\) subsystem code can be obtained from the \(\mathbb{Z}_q\) square-lattice surface code by gauging out the anyon \(e^{-1} m\) and applying single-qubit Clifford gates [23; Sec. 7.3]. During this process, the square lattice is effectively expanded to a honeycomb tiling [23; Fig. 12].
Primary Hierarchy
Abelian TQD stabilizer codeLattice stabilizer QLDPC Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum
Parents
Modular-qudit surface code Hamiltonians admit topological phases associated with \(\mathbb{Z}_q\) topological order [2].
Generalized homological-product CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum
Modular-qudit surface code
Children
The modular-qudit surface code for \(q=2\) reduces to the surface code.
The qudit Shor code is a small qudit surface code on a Möbius strip with smooth boundary, which is obtained from removing a face of the tesselation of the projective plane \(\mathbb{R}P^2\) [24; Fig. 4].
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Page edit log
- Victor V. Albert (2022-11-02) — most recent
- Victor V. Albert (2022-01-05)
- Ian Teixeira (2021-12-19)
Cite as:
“Modular-qudit surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qudit_surface