Here is a list of 2D stabilizer codes.

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Code Description
2D bosonization code A mapping between a 2D lattice quadratic Hamiltonian of Majorana modes and a 2D lattice of qubits. The original exact 2D bosonization code [1] is a stabilizer code whose generators are products of plaquettes and stars of the surface code, with gauge constraints that project onto a toric-code-like subspace with emergent fermions [1,2]. Finite-depth generalized local unitary Clifford circuits generate a family of equivalent local encodings with qubit-to-fermion ratio \(r = 1 + \frac{1}{2k}\) for any positive integer \(k\); the square-lattice compact encoding with \(r=1.5\) and the super-compact encoding with \(r=1.25\) are explicit examples [2].
2D color code Color code defined on a two-dimensional planar graph. Each face hosts two stabilizer generators, a Pauli-\(X\) and a Pauli-\(Z\) string acting on all the qubits of the face.
2D lattice stabilizer code Lattice stabilizer code in two Euclidean dimensions, using either the ordinary block notion of locality or the fermionic/Majorana notion of locality.
Abelian TQD stabilizer code Modular-qudit stabilizer code whose codewords realize a 2D Abelian twisted-quantum-double topological order on composite-dimensional qudits. For every finite Abelian group \(G=\prod_i \mathbb{Z}_{N_i}\) and every product of Type-I and Type-II cocycles, there is a Pauli stabilizer Hamiltonian realizing the corresponding Abelian TQD [3]. Equivalently, these codes exhaust the 2D Abelian topological orders that admit gapped boundaries [3,4].
Abelian quantum-double stabilizer code Modular-qudit stabilizer code whose codewords realize 2D modular gapped Abelian topological order with trivial cocycle. The corresponding anyon theory is defined by an Abelian group. The \(G=\mathbb{Z}_2\) instance on a torus is the toric code, and cyclic-group instances reduce to modular-qudit surface codes. All such codes can be realized by a stack of modular-qudit surface codes because all finite Abelian groups are direct products of cyclic groups.
Analog surface code An analog CSS version of the Kitaev surface code realizing a phase of 2D \(\mathbb{R}\) gauge theory.
BB5 code A BB code with weight-five stabilizer generators (contrasting with the weight-six checks of standard BB codes), designed and benchmarked for long chains of trapped ions [5].
Ball-Verstraete-Cirac (BVC) code A 2D fermion-into-qubit encoding that builds upon the JW transformation by eliminating the weight-\(O(n)\) non-local \(Z\)-type string at the expense of introducing an auxiliary qubit per site and local gauge constraints. See [2; Sec. IV.B] for details.
Bivariate bicycle (BB) code One of several Abelian 2BGA codes which admit time-optimal syndrome measurement circuits that can be implemented in a two-layer architecture, a generalization of the square-lattice architecture optimal for the surface codes. Codes can be classified by the weight of their checks, e.g., by BB\(w\) where \(w\) is the check weight.
Bravyi-Kitaev superfast (BKSF) code A single-error-detecting fermion-into-qubit encoding defined on a 2D qubit lattice whose stabilizers are associated with loops in the lattice. For the square-lattice edge ordering used in Ref. [2], the BKSF logical operators coincide with exact 2D bosonization on the dual lattice after relabeling \(X\) and \(Y\). The code can be generalized to a single error-correcting code (i.e., with distance three) on graphs of degree \(\geq 6\) [6].
Clifford-deformed surface code (CDSC) A generally non-CSS derivative of the surface code defined by applying a translationally invariant constant-depth Clifford circuit to the original (CSS) surface code. Unlike the surface code, CDSCs include codes whose thresholds and subthreshold performance are enhanced under noise biased towards dephasing. Examples of CDSCs include the XY code, XZZX code, and random CDSCs.
Compactified \(\mathbb{R}\) gauge theory code An integer-homology bosonic CSS code realizing 2D \(U(1)\) gauge theory on bosonic modes. The code can be obtained from the analog surface code by condensing certain anyons [7]. This results in a pinning of each mode to the space of periodic functions, which is the Hilbert space of a physical rotor, and can be thought of as compactification of the 2D \(\mathbb{R}\) gauge theory phase realized by the analog surface code.
Derby-Klassen (DK) code A fermion-into-qubit code defined on regular tilings with maximum degree 4 whose stabilizers are associated with loops in the tiling. The code outperforms several other encodings in terms of encoding rate [8; Table I]. It has been extended for models with several modes per site [9].
Double-semion stabilizer code A 2D lattice modular-qudit stabilizer code with qudit dimension \(q=4\) that realizes the 2D double semion topological phase. The code can be obtained from a \(\mathbb{Z}_4\) toric-code ground state by condensing the emergent boson \(e^2 m^2\); in the stabilizer construction this condensation is implemented by two-body measurements [3,10]. Its ground-state subspace can be mapped to that of the double-semion string-net model by a finite-depth quantum circuit with ancillas [3].
GKP-surface code A concatenated code whose outer code is a GKP code and whose inner code is a surface code, including toric surface-code variants [11,12], rotated surface codes [13–16], and XZZX surface codes [17].
Galois-qudit color code Extension of the color code to 2D lattices of Galois qudits.
Galois-qudit surface code Extension of the surface code to 2D lattices of Galois qudits.
Honeycomb (6.6.6) color code 2D color code defined on a (typically triangular) patch of the 6.6.6 (honeycomb) tiling. The usual triangular patch has three differently colored boundaries, encodes one logical qubit, and is local-Clifford equivalent to a folded surface/toric code with two smooth and two rough boundaries [18].
Kitaev surface code A family of Abelian topological CSS stabilizer codes whose generators are few-body \(X\)-type and \(Z\)-type Pauli strings associated to the stars and plaquettes, respectively, of a cellulation of a two-dimensional surface (with a qubit located at each edge of the cellulation). Codewords correspond to ground states of the surface code Hamiltonian, and error operators create or annihilate pairs of anyonic charges or vortices.
Klein-bottle surface code A family of Kitaev surface codes on the non-orientable Klein bottle.
Majorana color code A fermionic analogue of a 2D color code.
Majorana loop stabilizer code (MLSC) A single error-correcting fermion-into-qubit encoding defined on a 2D qubit lattice whose stabilizers are associated with loops in the lattice.
Majorana surface code Fermionic analogue of the surface code defined on a three-colorable 2D tiling whose face operators are non-overlapping even-Majorana stabilizers. Open patches with four or six alternating colored boundaries encode logical tetrons or hexons. The uniform 4.8.8, 6.6.6, and 4.6.12 tilings yield families with tetron, hexon, or dodecon building blocks and with twist-based lattice surgery supporting minimal-overhead logical Clifford gates [19].
Matching code Member of a class of qubit stabilizer codes based on the Abelian phase of the Kitaev honeycomb model.
Modular-qudit surface code Extension of the surface code to prime-dimensional [20,21] and more general modular qudits. 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.
Projective-plane surface code A family of Kitaev surface codes on the non-orientable 2-dimensional compact manifold \(\mathbb{R}P^2\) (in contrast to a genus-\(g\) surface). Whereas genus-\(g\) surface codes require \(2g\) logical qubits, qubit codes on \(\mathbb{R}P^2\) are made from a single logical qubit.
Rotated surface code Variant of the surface code defined on a square lattice that has been rotated 45 degrees such that qubits are on vertices, and both \(X\)- and \(Z\)-type check operators occupy plaquettes in an alternating checkerboard pattern.
Square-lattice cluster-state code A code based on the cluster state on a square lattice that was used in the first proposal for MBQC [22,23]. In the one-way model, the pre-entangled square-lattice cluster is a universal resource, and the computation is carried out entirely by adaptive single-qubit measurements.
Square-octagon (4.8.8) color code 2D color code defined on a patch of the 4.8.8 (square-octagon) tiling, which itself is obtained by applying a fattening procedure to the square lattice [24]. An equivalent description uses the Tetrakis square tiling (a.k.a. the Union Jack lattice), which is dual to the 4.8.8 lattice [25]. Among the three semiregular triangular 2D color-code families, the 4.8.8 family uses the fewest physical qubits for a given distance and is the only one of the three with transversal implementations of the full Clifford group [26].
Stellated color code A non-CSS color-code family on a lattice patch with a single central puncture that hosts a twist defect connected to the boundary by a domain wall.
Stellated surface code A twist-defect surface-code family parameterized by a rotational symmetry order \(s\), with a central toric-code twist connected to the boundary by a domain wall. The \(s=3\) member is the triangular surface code [27].
Super-compact fermion-to-qubit code A 2D fermion-into-qubit encoding on the square lattice obtained from exact 2D bosonization by a finite-depth generalized local unitary Clifford circuit, followed by re-pairing of Majorana modes and a slight lattice deformation. The code uses \(1.25\) qubits per fermion, improving on the square-lattice compact encoding with ratio \(r=1.5\). Its fermion-parity, hopping, and stabilizer operators have weights \(1\)-\(2\), \(2\)-\(6\), and \(12\), respectively [2; Table I].
Toric code Version of the Kitaev surface code on a square lattice with periodic boundary conditions, encoding two logical qubits. Being the first manifestation of the surface code, “toric code” is often an alternative name for the general construction. Twisted toric code [28; Fig. 8] refers to the construction on a torus with twisted (a.k.a. shifted) boundary conditions. In the original Hamiltonian construction, open Pauli-\(Z\) and Pauli-\(X\) strings create pairs of electric charges and magnetic vortices, and braiding one type around the other yields the nontrivial Abelian anyonic phase [20].
Triangular surface code A member of a twist-defect surface code family with a single central twist whose planar layout fits within a triangle. Triangle codes can be viewed as three conjoined surface-code patches projected into two dimensions, with weight-four plaquette stabilizers and weight-two edge stabilizers [27]. Symmetric distance-\(d\) triangle codes use \(3d^2/4+1/4\) data qubits, i.e., about \(25\%\) fewer than the rotated surface code for a given odd distance. Logical \(\overline{X}\), \(\overline{Y}\), and \(\overline{Z}\) operators can be supported on the three sides of the triangle, enabling initialization and measurement in any Pauli basis.
Truncated trihexagonal (4.6.12) color code 2D color code defined on a (typically triangular) patch of the 4.6.12 (truncated trihexagonal or square-hexagon-dodecagon) tiling.
Twist-defect color code A non-CSS extension of the 2D color code whose non-CSS stabilizer generators are associated with twist defects of the associated lattice. These twists terminate domain walls that permute color labels, Pauli labels, or interchange the two.
Twist-defect surface code A non-CSS extension of the 2D surface-code construction whose non-CSS stabilizer generators are associated with twist defects of the associated lattice. A related construction [29] doubles the number of qubits in the lattice via symplectic doubling.
Twisted XZZX toric code A cyclic code that can be thought of as the XZZX toric code with shifted (a.k.a twisted) boundary conditions. Admits a set of stabilizer generators that are cyclic shifts of a particular weight-four \(XZZX\) Pauli string. For example, a seven-qubit \([[7,1,3]]\) variant has stabilizers generated by cyclic shifts of \(XZIZXII\) [30]. Codes encode either one or two logical qubits, depending on qubit geometry, and perform well against biased noise [31]. See Ref. [29] for a table of some of these for small instances, where they are called genus-one genon codes.
XY surface code Non-CSS derivative of the surface code whose generators are \(XXXX\) and \(YYYY\), obtained by mapping \(Z \to Y\) in the surface code.
XYZ color code Non-CSS variant of the 6.6.6 color code whose generators are \(XZXZXZ\) and \(ZYZYZY\) Pauli strings associated to each hexagonal in the hexagonal (6.6.6) tiling. A further variation called the domain wall color code admits generators of the form \(XXXZZZ\) and \(ZZZXXX\) [32].
XYZ\(^2\) hexagonal stabilizer code An instance of the matching code based on the Kitaev honeycomb model. It is described on a honeycomb tiling with \(XYZXYZ\) stabilizers on each hexagonal plaquette. Each vertical pair of qubits has an \(XX\), \(YY\), or \(ZZ\) link stabilizer depending on the orientation of the plaquette stabilizers.
XZZX surface code Non-CSS variant of the rotated surface code whose generators are \(XZZX\) Pauli strings associated, clock-wise, to the vertices of each face of a two-dimensional lattice (with a qubit located at each vertex of the tessellation).
\(U(1)_{2n} \times U(1)_{-2m}\) Chern-Simons GKP code A non-CSS multimode GKP code defined on a 2D mode lattice that encodes a qudit logical space and whose excitations are characterized by the \(U(1)_{2n} \times U(1)_{-2m}\) Chern-Simons theory. The code can be obtained from the analog surface code by condensing certain anyons [7].
\([[108,8,10]]\) BB6 code A bivariate bicycle (BB) code with parameters \([[108,8,10]]\) and weight-six stabilizer generators [33].
\([[13,1,5]]\) twisted toric code Thirteen-qubit twisted toric code for which there is a set of stabilizer generators consisting of cyclic permutations of the \(XZZX\)-type Pauli string \(XIZZIXIIIIIII\). The code can be thought of as a small twisted XZZX code [34; Exam. 11 and Fig. 3].
\([[14,3,3]]\) Rhombic dodecahedron surface code A \([[14,3,3]]\) twist-defect surface code whose qubits lie on the vertices of a rhombic dodecahedron. Its non-CSS nature is due to twist defects [35] stemming from the geometry of the polytope. A local-Clifford-equivalent clean realization has only \(X\)- and \(Z\)-type operators on its four-valent vertices, and its symplectic double is a \([[28,6,3]]\) genus-three code [29].
\([[144,12,12]]\) gross code A BB code which requires less physical and ancilla qubits (for syndrome extraction) than the surface code with the same number of logical qubits and distance. The gross code is equivalent to 8 copies of the surface code via a constant-depth Clifford circuit, and is an element of a larger family of 2D stabilizer codes [36]. The name stems from the fact that a gross is a dozen dozen.
\([[17,1,5]]\) 4.8.8 color code Seventeen-qubit doubly even 2D color code that admits a transversal implementation of the logical Clifford group. The smallest distance-five CSS code has \(n=17\) [37]. It is also a normal self-dual CSS code whose transversal Hadamard acts logically, making it suitable as an inner code for fifth-order magic-state distillation [38].
\([[288,12,18]]\) double-gross code A bivariate bicycle (BB) code with parameters \([[288,12,18]]\) and weight-six stabilizer generators [33].
\([[4,1,2]]\) Leung-Nielsen-Chuang-Yamamoto (LNCY) code A four-qubit CSS stabilizer code that is the only qubit CSS code with such parameters.
\([[4,1,2]]\) twist-defect code A four-qubit non-CSS stabilizer code that can be interpreted as the smallest triangular color code with \(x\)-, \(y\)-, and \(z\)-type Pauli boundaries [39; Fig. 7], and equivalently as a small twist-defect surface code on a tetrahedron inscribed in a sphere [29]. It is the only non-CSS qubit stabilizer code with parameters \([[4,1,2]]\). The code admits weight-three stabilizer generators and weight-two logical Pauli \(X,Y,Z\) operators.
\([[4,2,2]]\) Four-qubit code A four-qubit hyperbolic self-dual CSS stabilizer code that is the smallest two-logical-qubit stabilizer code to detect a single-qubit error. It is unique for its parameters [40; Thm. 8].
\([[5,1,2]]\) rotated surface code A rotated surface code on one rung of a ladder, with one qubit on the rung, and four qubits surrounding it.
\([[5,1,3]]\) Five-qubit perfect code Five-qubit cyclic stabilizer code that is the smallest qubit stabilizer code to correct a single-qubit error.
\([[6,2,2]]\) \(C_6\) code Error-detecting normal self-dual CSS code on three qubit pairs that encodes a logical qubit pair and detects any error acting on one pair [41]. In Knill’s \(C_4/C_6\) architecture, this code is used at the second and higher concatenation levels.
\([[6,4,2]]\) error-detecting code Self-complementary six-qubit code with rate \(2/3\) that is unique for its parameters, up to equivalence [42; Tab. III]. Concatenations of this code with itself yield the \([[6^r,4^r,2^r]]\) level-\(r\) many-hypercube code [43].
\([[7,1,3]]\) Steane code A \([[7,1,3]]\) self-dual CSS code that is the smallest qubit CSS code to correct a single-qubit error [44]. The code is constructed using the classical binary \([7,4,3]\) Hamming code for protecting against both \(X\) and \(Z\) errors.
\([[7,1,3]]\) twist-defect surface code A \([[7,1,3]]\) code (different from the Steane code) that is a small example of a twist-defect surface code.
\([[72,12,6]]\) BB6 code A bivariate bicycle (BB) code with parameters \([[72,12,6]]\) and weight-six stabilizer generators [33].
\([[8,3,2]]\) Surface code on a cube An \([[8,3,2]]\) twist-defect surface code whose qubits lie on the vertices of a cube. It is obtained by three-coloring the faces of a cube and placing \(X\), \(Y\), and \(Z\) stabilizer generators on each pair of faces of the same color. Its non-CSS nature is due to twist defects [35] stemming from the geometry of the polytope.
\([[9,1,3]]\) Shor code Nine-qubit CSS code that is the first quantum error-correcting code [45; ID 8802]. Among indecomposable \([[9,1,3]]\) CSS codes, the Shor code has the largest automorphism group [46].
\([[9,1,3]]\) Surface-17 code A \([[9,1,3]]\) rotated surface code named for the sum of its 9 data qubits and 8 syndrome qubits. It is one of the four inequivalent CSS gauge fixings of the nine-qubit Bacon-Shor code [46]. It uses the smallest number of qubits to perform fault-tolerant error correction on a surface code with parallel syndrome extraction.
\([[90,8,10]]\) BB6 code A bivariate bicycle (BB) code with parameters \([[90,8,10]]\) and weight-six stabilizer generators [33].

References

[1]
Y.-A. Chen, A. Kapustin, and Đ. Radičević, “Exact bosonization in two spatial dimensions and a new class of lattice gauge theories”, Annals of Physics 393, 234 (2018) arXiv:1711.00515 DOI
[2]
Y.-A. Chen and Y. Xu, “Equivalence between Fermion-to-Qubit Mappings in two Spatial Dimensions”, PRX Quantum 4, (2023) arXiv:2201.05153 DOI
[3]
T. D. Ellison, Y.-A. Chen, A. Dua, W. Shirley, N. Tantivasadakarn, and D. J. Williamson, “Pauli Stabilizer Models of Twisted Quantum Doubles”, PRX Quantum 3, (2022) arXiv:2112.11394 DOI
[4]
J. Kaidi, Z. Komargodski, K. Ohmori, S. Seifnashri, and S.-H. Shao, “Higher central charges and topological boundaries in 2+1-dimensional TQFTs”, SciPost Physics 13, (2022) arXiv:2107.13091 DOI
[5]
M. Ye and N. Delfosse, “Quantum error correction for long chains of trapped ions”, Quantum 9, 1920 (2025) arXiv:2503.22071 DOI
[6]
K. Setia, S. Bravyi, A. Mezzacapo, and J. D. Whitfield, “Superfast encodings for fermionic quantum simulation”, Physical Review Research 1, (2019) arXiv:1810.05274 DOI
[7]
J. C. Magdalena de la Fuente, T. D. Ellison, M. Cheng, and D. J. Williamson, “Topological Stabilizer Models on Continuous Variables”, Physical Review X 16, (2026) arXiv:2411.04993 DOI
[8]
C. Derby, J. Klassen, J. Bausch, and T. Cubitt, “Compact fermion to qubit mappings”, Physical Review B 104, (2021) arXiv:2003.06939 DOI
[9]
L. Clinton, T. Cubitt, B. Flynn, F. M. Gambetta, J. Klassen, A. Montanaro, S. Piddock, R. A. Santos, and E. Sheridan, “Towards near-term quantum simulation of materials”, Nature Communications 15, (2024) arXiv:2205.15256 DOI
[10]
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
[11]
K. Fukui, A. Tomita, A. Okamoto, and K. Fujii, “High-Threshold Fault-Tolerant Quantum Computation with Analog Quantum Error Correction”, Physical Review X 8, (2018) arXiv:1712.00294 DOI
[12]
C. Vuillot, H. Asasi, Y. Wang, L. P. Pryadko, and B. M. Terhal, “Quantum error correction with the toric Gottesman-Kitaev-Preskill code”, Physical Review A 99, (2019) arXiv:1810.00047 DOI
[13]
K. Noh and C. Chamberland, “Fault-tolerant bosonic quantum error correction with the surface–Gottesman-Kitaev-Preskill code”, Physical Review A 101, (2020) arXiv:1908.03579 DOI
[14]
M. V. Larsen, C. Chamberland, K. Noh, J. S. Neergaard-Nielsen, and U. L. Andersen, “Fault-Tolerant Continuous-Variable Measurement-based Quantum Computation Architecture”, PRX Quantum 2, (2021) arXiv:2101.03014 DOI
[15]
K. Noh, C. Chamberland, and F. G. S. L. Brandão, “Low-Overhead Fault-Tolerant Quantum Error Correction with the Surface-GKP Code”, PRX Quantum 3, (2022) arXiv:2103.06994 DOI
[16]
M. Lin, C. Chamberland, and K. Noh, “Closest Lattice Point Decoding for Multimode Gottesman-Kitaev-Preskill Codes”, PRX Quantum 4, (2023) arXiv:2303.04702 DOI
[17]
J. Zhang, Y.-C. Wu, and G.-P. Guo, “Concatenation of the Gottesman-Kitaev-Preskill code with the XZZX surface code”, Physical Review A 107, (2023) arXiv:2207.04383 DOI
[18]
A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
[19]
D. Litinski and F. von Oppen, “Quantum computing with Majorana fermion codes”, Physical Review B 97, (2018) arXiv:1801.08143 DOI
[20]
A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons”, Annals of Physics 303, 2 (2003) arXiv:quant-ph/9707021 DOI
[21]
S. S. Bullock and G. K. Brennen, “Qudit surface codes and gauge theory with finite cyclic groups”, Journal of Physics A: Mathematical and Theoretical 40, 3481 (2007) arXiv:quant-ph/0609070 DOI
[22]
R. Raussendorf, D. Browne, and H. Briegel, “The one-way quantum computer–a non-network model of quantum computation”, Journal of Modern Optics 49, 1299 (2002) arXiv:quant-ph/0108118 DOI
[23]
R. Raussendorf and H. J. Briegel, “A One-Way Quantum Computer”, Physical Review Letters 86, 5188 (2001) DOI
[24]
H. Bombin and M. A. Martin-Delgado, “Exact topological quantum order inD=3and beyond: Branyons and brane-net condensates”, Physical Review B 75, (2007) arXiv:cond-mat/0607736 DOI
[25]
H. G. Katzgraber, H. Bombin, R. S. Andrist, and M. A. Martin-Delgado, “Topological color codes on Union Jack lattices: a stable implementation of the whole Clifford group”, Physical Review A 81, (2010) arXiv:0910.0573 DOI
[26]
A. J. Landahl, J. T. Anderson, and P. R. Rice, “Fault-tolerant quantum computing with color codes”, (2011) arXiv:1108.5738
[27]
T. J. Yoder and I. H. Kim, “The surface code with a twist”, Quantum 1, 2 (2017) arXiv:1612.04795 DOI
[28]
N. P. Breuckmann and J. N. Eberhardt, “Balanced Product Quantum Codes”, IEEE Transactions on Information Theory 67, 6653 (2021) arXiv:2012.09271 DOI
[29]
S. Burton, E. Durso-Sabina, and N. C. Brown, “Genons, Double Covers and Fault-tolerant Clifford Gates”, (2024) arXiv:2406.09951
[30]
A. Robertson, C. Granade, S. D. Bartlett, and S. T. Flammia, “Tailored Codes for Small Quantum Memories”, Physical Review Applied 8, (2017) arXiv:1703.08179 DOI
[31]
Q. Xu, N. Mannucci, A. Seif, A. Kubica, S. T. Flammia, and L. Jiang, “Tailored XZZX codes for biased noise”, Physical Review Research 5, (2023) arXiv:2203.16486 DOI
[32]
K. Tiurev, A. Pesah, P.-J. H. S. Derks, J. Roffe, J. Eisert, M. S. Kesselring, and J.-M. Reiner, “Domain Wall Color Code”, Physical Review Letters 133, (2024) arXiv:2307.00054 DOI
[33]
S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, “High-threshold and low-overhead fault-tolerant quantum memory”, Nature 627, 778 (2024) arXiv:2308.07915 DOI
[34]
A. A. Kovalev, I. Dumer, and L. P. Pryadko, “Design of additive quantum codes via the code-word-stabilized framework”, Physical Review A 84, (2011) arXiv:1108.5490 DOI
[35]
H. Bombin, “Topological Order with a Twist: Ising Anyons from an Abelian Model”, Physical Review Letters 105, (2010) arXiv:1004.1838 DOI
[36]
Z. Liang, B. Yang, J. T. Iosue, and Y.-A. Chen, “Operator algebra and algorithmic construction of boundaries and defects in (2+1)D topological Pauli stabilizer codes”, (2025) arXiv:2410.11942
[37]
M. F. Ezerman, S. Jitman, S. Ling, and D. V. Pasechnik, “CSS-Like Constructions of Asymmetric Quantum Codes”, IEEE Transactions on Information Theory 59, 6732 (2013) arXiv:1207.6512 DOI
[38]
J. Haah, M. B. Hastings, D. Poulin, and D. Wecker, “Magic state distillation with low space overhead and optimal asymptotic input count”, Quantum 1, 31 (2017) arXiv:1703.07847 DOI
[39]
M. S. Kesselring, F. Pastawski, J. Eisert, and B. J. Brown, “The boundaries and twist defects of the color code and their applications to topological quantum computation”, Quantum 2, 101 (2018) arXiv:1806.02820 DOI
[40]
E. M. Rains, “Quantum codes of minimum distance two”, (1997) arXiv:quant-ph/9704043
[41]
E. Knill, “Quantum computing with realistically noisy devices”, Nature 434, 39 (2005) arXiv:quant-ph/0410199 DOI
[42]
A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum Error Correction via Codes over GF(4)”, (1997) arXiv:quant-ph/9608006
[43]
H. Goto, “High-performance fault-tolerant quantum computing with many-hypercube codes”, Science Advances 10, (2024) arXiv:2403.16054 DOI
[44]
B. Shaw, M. M. Wilde, O. Oreshkov, I. Kremsky, and D. A. Lidar, “Encoding one logical qubit into six physical qubits”, Physical Review A 78, (2008) arXiv:0803.1495 DOI
[45]
Qiskit Community. Qiskit QEC framework. https://github.com/qiskit-community/qiskit-qec
[46]
A. Cross and D. Vandeth, “Small Binary Stabilizer Subsystem Codes”, (2025) arXiv:2501.17447