Here is a list of non-qubit quantum subsystem codes. For operator-algebra qubit codes, see Operator-algebra qubit codes.
| Code | Description |
|---|---|
| Chiral semion subsystem code | Modular-qudit subsystem stabilizer code with qudit dimension \(q=4\) that is characterized by the chiral semion topological phase. Admits a set of geometrically local stabilizer generators on a torus. |
| Lattice subsystem code | A geometrically local qubit, modular-qudit, or Galois-qudit subsystem stabilizer code with qudits organized on a lattice modeled by the additive group \(\mathbb{Z}^D\) for spatial dimension \(D\), using either the ordinary block notion of locality or the fermionic/Majorana notion of locality. On an infinite lattice, its gauge group is generated by few-site Pauli operators and their translations, in which case the code is called translationally invariant subsystem code. The stabilizer group may contain generators of unbounded weight, distinguishing these codes from stabilizer codes with bounded-weight generators for which some logical qubits were re-assigned to be gauge qubits. |
| Modular-qudit subsystem color code | An extension of subsystem color codes to modular qudits. Codes are defined analogously to qubit subsystem color codes, but a directionality is required in order to make the modular-qudit stabilizers commute [1; Sec. VII]. |
| QLDPC subsystem code | Member of a family of subsystem stabilizer codes for which the number of sites participating in each gauge generator and the number of gauge generators that each site participates in are both bounded by a constant as \(n\to\infty\). The stabilizer group may contain generators of unbounded weight, distinguishing these codes from stabilizer codes with bounded-weight generators for which some logical qubits were re-assigned to be gauge qubits. |
| Subsystem CSS code | A subsystem stabilizer code admitting a set of gauge-group generators that are either \(Z\)-type or \(X\)-type operators. This ensures that the associated stabilizer group is also CSS. |
| Subsystem Galois-qudit CSS code | Galois-qudit subsystem stabilizer code which admits a set of gauge-group generators which consist of either all-\(Z\) or all-\(X\) Galois-qudit Pauli strings. |
| Subsystem Galois-qudit code | Subsystem QECC encoding into a \(q^n\)-dimensional Hilbert space consisting of \(n\) Galois qudits. |
| Subsystem Galois-qudit stabilizer code | Galois-qudit generalization of a subsystem qubit stabilizer code. Can be obtained by taking a Galois-qudit stabilizer code and assigning some of its logical qudits to be gauge qudits. |
| Subsystem QECC | A quantum code which encodes quantum information in a tensor factor of a subspace that is decomposed into a tensor product of subsystems. |
| Subsystem group-based quantum code | Group-based quantum code whose codespace admits a tensor-product decomposition into logical and gauge factors. |
| Subsystem modular-qudit CSS code | Modular-qudit subsystem stabilizer code which admits a set of gauge-group generators which consist of either all-\(Z\) or all-\(X\) modular-qudit Pauli strings. This ensures that the code’s stabilizer group is also CSS. |
| Subsystem modular-qudit code | Subsystem QECC encoding into a \(q^n\)-dimensional Hilbert space consisting of \(n\) modular qudits. |
| Subsystem modular-qudit stabilizer code | Modular-qudit generalization of a subsystem qubit stabilizer code. Can be obtained by taking a modular-qudit stabilizer code and assigning some of its logical qudits to be gauge qudits. For composite qudit dimensions, such codes need not encode an integer number of qudits. |
| Subsystem stabilizer code | A subsystem code that is derived from a stabilizer code by assigning some factors of the stabilizer code’s logical tensor-product structure to gauge degrees of freedom. |
| \(\mathbb{Z}_3\times\mathbb{Z}_9\)-fusion subsystem code | Modular-qudit 2D subsystem stabilizer code whose low-energy excitations realize a non-modular anyon theory with \(\mathbb{Z}_3\times\mathbb{Z}_9\) fusion rules. Encodes two qutrits when put on a torus. |
| \(\mathbb{Z}_q^{(1)}\) subsystem code | Modular-qudit subsystem code, based on the Kitaev honeycomb model [2] and its generalization [3], that is characterized by the \(\mathbb{Z}_q^{(1)}\) anyon theory [4], which is modular for odd prime \(q\) and non-modular otherwise. Encodes a single \(q\)-dimensional qudit when put on a torus for odd \(q\), and a \(q/2\)-dimensional qudit for even \(q\). This code can be constructed using geometrically local gauge generators, but does not admit geometrically local stabilizer generators. For \(q=2\), the code reduces to the subsystem code underlying the Kitaev honeycomb model code as well as the honeycomb Floquet code. |
References
- [1]
- F. H. E. Watson, E. T. Campbell, H. Anwar, and D. E. Browne, “Qudit color codes and gauge color codes in all spatial dimensions”, Physical Review A 92, (2015) arXiv:1503.08800 DOI
- [2]
- A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006) arXiv:cond-mat/0506438 DOI
- [3]
- M. Barkeshli, H.-C. Jiang, R. Thomale, and X.-L. Qi, “Generalized Kitaev Models and Extrinsic Non-Abelian Twist Defects”, Physical Review Letters 114, (2015) arXiv:1405.1780 DOI
- [4]
- P. H. Bonderson, Non-Abelian Anyons and Interferometry, California Institute of Technology, 2007 DOI