Here is a list of quantum subsystem codes.

[Jump to code graph excerpt]

Code Description
2D subsystem color code A subsystem version of the 2D color code.
3D Bacon-Shor code Generalization of the Bacon-Shor code to three dimensions that was conjectured to be a self-correcting memory. It is defined on a cubic lattice and admits sheet-like stabilizer generators.
3D Kitaev honeycomb code 3D subsystem stabilizer code whose Hamiltonian is a 3D generalization of the Kitaev honeycomb model. One of the phases realized by the 3D Kitaev honeycomb Hamiltonian is that of the 3D fermionic surface code [1].
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.
Bravyi-Bacon-Shor (BBS) code An \([[n,k,d]]\) CSS subsystem stabilizer code generalizing Bacon-Shor codes to a larger set of qubit geometries. Defined through a binary matrix \(A\) such that physical qubits live on sites \((i,j)\) whenever \(A_{i,j}=1\). The gauge group is generated by 2-qubit operators, including \(XX\) interations between any two qubits sharing a column in \(A\), and \(ZZ\) interations between two qubits sharing a row. The code parameters are: \(n=\sum_{i,j}A_{i,j}\), \(k=\text{rank}(A)\), and the distance is the minimum weight of any row or column.
CSS-Plaquette code Generalization of the Bacon-Shor code to three dimensions, defined on a cubic lattice and admitting string-like stabilizer generators.
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 [2–4] 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 [5], 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 [6,7] of a code defined on a three-valent hypergraph associated with the five-squares lattice [8].
Heavy-hexagon code Subsystem stabilizer code on the heavy-hexagonal point set 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 point set 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 [9].
Holographic hybrid code Holographic tensor-network code constructed out of alternating isometries of the five-qubit and \([[4,1,1,2]]\) Bacon-Shor codes.
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 [10]. Ising anyons also exist in other phases, such as the fractional quantum Hall phase [11].
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.
Majorana subsystem stabilizer code A Majorana stabilizer code with some of its logical qubits denoted as gauge qubits and not used for storage of logical information.
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 [12; Sec. VII].
Sarvepalli-Brown subsystem code Member of a family of non-CSS subsystem codes constructed from hypergraphs that satisfy certain assumptions [6; Construction C].
Sparse subsystem code A geometrically local qubit, modular-qudit, or Galois-qudit subsystem stabilizer code for which the number of sites participating in each gauge-group generator and the number of gauge-group generators that each site participates in are both bounded by a constant as \(n\to\infty\).
Subsystem CSS code Subsystem stabilizer code which admits a set of gauge-group generators which consist of either all-\(Z\) or all-\(X\) Pauli strings. This ensures that the code's 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 qubits to be gauge qubits.
Subsystem Hypergraph Product Simplex (SHYPS) code Family of quantum LDPC codes obtained by combining the subsystem hypgergraph product code construction with classical simplex codes. The result are CSS subsystem codes with weight-three gauge generators and code parameters \([n=(2^r − 1)^2, k=r^2, d=2^{r-1}]\) for \(r \geq 3\).
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 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 homological product code A CSS subsystem code constructed from a product of two (subspace) CSS codes. The case for qubits is formulated below, but these codes have also been extended to Galois qudits [13].
Subsystem hyperbolic surface code Subsystem generalization of the surface code on a 2D hyperbolic tesselation with gauge-group generators of weight at most three. An \(\{r,s\}\) hyperbolic tesselation with \(E\) edges yields a \([[3E/2,(1/2-2/r)E+2,(1-2/r)E,d]]\) subsystem code.
Subsystem hypergraph product (SHP) code A CSS subsystem version of the generalized Shor code that has the same parameters as the subspace version, but requires fewer stabilizer measurements, resulting in a simpler error recovery routine. The code can also be thought of as a subsystem version of an HGP code because two such codes reduce to an HGP code upon gauge fixing [14; Sec. III]. The code can be obtained from a generalized Shor code by removing certain stabilizers that do no affect the code distance.
Subsystem lifted-product (SLP) code Member of a family of subsystem CSS codes constructed from a subsystem hypergraph product of a lifted binary linear code.
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 qubits to be gauge qubits. For composite qudit dimensions, such codes need not encode an integer number of qudits.
Subsystem qubit code Subsystem QECC encoding into a \(2^n\)-dimensional (i.e., \(n\)-qubit) Hilbert space.
Subsystem qubit stabilizer code A stabilizer code with some of its logical qubits denoted as gauge qubits and not used for storage of logical information. Note that this doesn't lead to new codes but does lead to new error correction and fault tolerance procedures. Subsystem codes are denoted by \([[n,k,g,d]]\), similar to stabilizer codes, but with an extra parameter \(g\) denoting the number of gauge qubits.
Subsystem rotated surface code Subsystem version of the rotated surface code.
Subsystem spacetime circuit code Subsystem stabilizer code obtained from a spacetime circuit code by gauging out logical operators that correspond to circuit faults with trivial effect [15; Sec. 5.4].
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 [16–18]. One version uses two qubits at each site [19], while other manifestations utilize a single qubit per site and only weight-two (two-body) interactions [17,20]. 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 [21]. See Ref. [21] for its gauge generators.
\([[9,1,4,3]]\) 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 [10] and its generalization [22], that is characterized by the \(\mathbb{Z}_q^{(1)}\) anyon theory [23], 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]
S. Mandal and N. Surendran, “Exactly solvable Kitaev model in three dimensions”, Physical Review B 79, (2009) arXiv:0801.0229 DOI
[2]
K. I. Kugel’ and D. I. Khomskiĭ, “The Jahn-Teller effect and magnetism: transition metal compounds”, Soviet Physics Uspekhi 25, 231 (1982) DOI
[3]
J. Dorier, F. Becca, and F. Mila, “Quantum compass model on the square lattice”, Physical Review B 72, (2005) arXiv:cond-mat/0501708 DOI
[4]
Z. Nussinov and J. van den Brink, “Compass and Kitaev models -- Theory and Physical Motivations”, (2013) arXiv:1303.5922
[5]
K. Betsumiya and A. Munemasa, “On triply even binary codes”, Journal of the London Mathematical Society 86, 1 (2012) arXiv:1012.4134 DOI
[6]
P. Sarvepalli and K. R. Brown, “Topological subsystem codes from graphs and hypergraphs”, Physical Review A 86, (2012) arXiv:1207.0479 DOI
[7]
V. V. Gayatri and P. K. Sarvepalli, “Decoding Algorithms for Hypergraph Subsystem Codes and Generalized Subsystem Surface Codes”, (2018) arXiv:1805.12542
[8]
M. Suchara, S. Bravyi, and B. Terhal, “Constructions and noise threshold of topological subsystem codes”, Journal of Physics A: Mathematical and Theoretical 44, 155301 (2011) arXiv:1012.0425 DOI
[9]
B. Hetényi and J. R. Wootton, “Creating Entangled Logical Qubits in the Heavy-Hex Lattice with Topological Codes”, PRX Quantum 5, (2024) arXiv:2404.15989 DOI
[10]
A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006) arXiv:cond-mat/0506438 DOI
[11]
S. Bravyi, “Universal quantum computation with theν=5∕2fractional quantum Hall state”, Physical Review A 73, (2006) arXiv:quant-ph/0511178 DOI
[12]
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
[13]
W. Zeng and L. P. Pryadko, “Minimal distances for certain quantum product codes and tensor products of chain complexes”, Physical Review A 102, (2020) arXiv:2007.12152 DOI
[14]
M. Li and T. J. Yoder, “A Numerical Study of Bravyi-Bacon-Shor and Subsystem Hypergraph Product Codes”, (2020) arXiv:2002.06257
[15]
N. Delfosse and A. Paetznick, “Spacetime codes of Clifford circuits”, (2023) arXiv:2304.05943
[16]
E. Rowell, R. Stong, and Z. Wang, “On classification of modular tensor categories”, (2009) arXiv:0712.1377
[17]
H. Bombin, M. Kargarian, and M. A. Martin-Delgado, “Interacting anyonic fermions in a two-body color code model”, Physical Review B 80, (2009) arXiv:0811.0911 DOI
[18]
H. Bombin, G. Duclos-Cianci, and D. Poulin, “Universal topological phase of two-dimensional stabilizer codes”, New Journal of Physics 14, 073048 (2012) arXiv:1103.4606 DOI
[19]
T. D. Ellison, Y.-A. Chen, A. Dua, W. Shirley, N. Tantivasadakarn, and D. J. Williamson, “Pauli topological subsystem codes from Abelian anyon theories”, Quantum 7, 1137 (2023) arXiv:2211.03798 DOI
[20]
H. Bombin, “Topological subsystem codes”, Physical Review A 81, (2010) arXiv:0908.4246 DOI
[21]
Z. Jiang and E. G. Rieffel, “Non-commuting two-local Hamiltonians for quantum error suppression”, Quantum Information Processing 16, (2017) arXiv:1511.01997 DOI
[22]
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
[23]
P. H. Bonderson, Non-Abelian Anyons and Interferometry, California Institute of Technology, 2007 DOI