Here is a list of lattice subsystem stabilizer codes.
Code Description
2D subsystem color code A subsystem version of the 2D color code.
3D subsystem color code A subsystem version of the 3D color code.
3D subsystem surface code Subsystem generalization of the surface code on a 3D cubic lattice with gauge-group generators of weight at most three.
Bacon-Shor code Subsystem CSS code defined on an \(m_1 \times m_2\) lattice of qubits that generalizes the \([[9,1,3]]\) (subspace) Shor code. It is said to be symmetric when \(m_1=m_2\), and asymmetric otherwise.
Capped color code (CCC) A non-geometrically local subsystem color code consisting of two layers of 2D color code stacked together and topped (or capped) by a single qubit. Gauge fixing yields two types of codes, capped color codes in H or T form. Layers of 2D color codes can also be stacked together in a recursive construction, yielding recursive capped color codes (RCCCs).
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.
Compass code Subspace or subsystem CSS code defined by gauge-fixing the Bacon-Shor code, i.e., the code whose gauge group consists of terms in the compass model Hamiltonian [1–3] on a square lattice. Families of random codes perform well against biased noise and spatially dependent (i.e., asymmetric) noise.
Doubled color code Family of \([[2t^3+8t^2+6t-1,1,2t+1]]\) subsystem color codes (with \(t\geq 1\)), constructed using a generalization of the doubling transformation [4], that admit a Clifford + \(T\) transversal gate set using gauge fixing.
Generalized five-squares code Member of a family of subsystem codes that are generalizations [5,6] of a code defined on a three-valent hypergraph associated with the five-squares lattice [7].
Heavy-hexagon code Subsystem stabilizer code on the heavy-hexagonal lattice that combines Bacon-Shor and surface-code stabilizers. Encodes one logical qubit into \(n=(5d^2-2d-1)/2\) physical qubits with distance \(d\). The heavy-hexagonal lattice allows for low degree (at most 3) connectivity between all the data and ancilla qubits, which is suitable for fixed-frequency transom qubits subject to frequency collision errors. The code can be split into a surface and a Bacon-Shor code, with the idling qubits of one code serving as the physical qubits of the other [8].
Kitaev honeycomb code Code whose logical subspace is labeled by different fusion outcomes of Ising anyons present in the Ising-anyon topological phase of the Kitaev honeycomb model [9]. Each logical qubit is constructed out of four Majorana operators, which admit braiding-based gates due to their non-Abelian statistics and which can be used for topological quantum computation. Ising anyons also exist in other phases, such as the fractional quantum Hall phase [10].
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 \(D\). On an infinite lattice, its gauge and stabilizer groups are generated by few-site Pauli operators and their translations, in which case the code is called translationally invariant subsystem code. Boundary conditions have to be imposed on the lattice in order to obtain finite-dimensional versions, in which case the stabilizer group may no longer be generated by few-site Pauli operators. Lattice defects and boundaries between different codes can also be introduced. Lattice subsystem stabilizer code Hamiltonians described by an Abelian anyon theory do not always realize the corresponding anyonic topological order in their ground-state subspace and may exhibit a rich phase diagram.
Modular-qudit subsystem color code An extension of subsystem color codes to modular qudits. Codes are defined analogous to qubit subsystem color codes, but a directionality is required in order to make the modular-qudit stabilizer commute [11; Sec. VII].
Subsystem color code A subsystem version of the color code. One way to obtain it is by expanding the vertices of a two-colex embedded in a surface of genus \(g\). Vertex expansion consists of spl every vertex into a triangle and splitting every edge into a pair of edges.
Subsystem rotated surface code Subsystem version of the rotated surface code.
Subsystem surface code Subsystem version of the surface code defined on a square lattice with qubits placed at every vertex and center of everry edge.
Three-fermion (3F) subsystem code 2D subsystem stabilizer code whose low-energy excitations realize the three-fermion anyon theory [12–14]. One version uses two qubits at each site [15], while other manifestations utilize a single qubit per site and only weight-two (two-body) interactions [13,16]. All are expected to be equivalent to each other via a local constant-depth Clifford circuit.
Trapezoid subsystem code A member of a family of BBS codes with weight-two (two-body) gauge generators designed to suppress errors in adiabatic quantum computation.
\([[4,1,1,2]]\) Four-qubit subsystem code Error-detecting four-qubit subsystem stabilizer code encoding one logical qubit and one gauge qubit.
\([[6,2,3,2]]\) BBS code Error-detecting six-qubit BBS code that can suppress errors in adiabatic quantum computation [17]. See Ref. [17] for its gauge generators.
\([[9,1,3,3]]\) Nine-qubit Bacon-Shor code Error-correcting nine-qubit subsystem stabilizer code encoding one logical qubit and three gauge qubits.
\(\mathbb{Z}_3\times\mathbb{Z}_9\)-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 [9] and its generalization [18], that is characterized by the \(\mathbb{Z}_q^{(1)}\) anyon theory [19], 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.

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