Here is a list of color codes, with or without defects.
| Code | Description |
|---|---|
| 2D color code | Color code defined on a graph embedded in a two-dimensional surface. Each face hosts two stabilizer generators, a Pauli-\(X\) and a Pauli-\(Z\) string acting on all the qubits of the face. |
| 3D color code | Color code defined on a four-valent, four-colorable 3-colex in a 3-manifold. In the original colex realization, qubits sit on vertices, \(X\)-type stabilizers are attached to 3-cells, and \(Z\)-type stabilizers are attached to faces [1]. |
| Ball code | A distance-two color code defined on a colorable \(D\)-ball, equivalently on a \(D\)-colex with boundary [2; Appx. A]. In the morphing construction of Ref. [2], ball codes arise as the child codes associated with the morphed ball-like regions. This family includes hypercube codes (defined on balls constructed from hyperoctahedra) and 3D ball codes (defined on duals of certain Archimedean solids). |
| Color code | Member of a family of qubit CSS codes defined on particular \(D\)-dimensional graphs. |
| Cubic honeycomb color code | 3D color code defined on a four-colorable bitruncated cubic honeycomb uniform tiling. |
| 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 [3]. |
| Hyperbolic color code | An extension of the color code construction to hyperbolic manifolds. As opposed to there being only three types of uniform three-valent and three-colorable lattice tilings in the 2D Euclidean plane, there is an infinite number of admissible hyperbolic tilings in the 2D hyperbolic plane [4]. Certain double covers of hyperbolic tilings also yield admissible tilings [5]. Other admissible hyperbolic tilings can be obtained via a fattening procedure [1]; see also a construction based on the more general quantum pin codes [6]. |
| Quasi-hyperbolic color code | An extension of the color code construction to quasi-hyperbolic 3-manifolds, e.g., a product of a 2D hyperbolic surface and a circle. |
| 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 [1]. An equivalent description uses the Tetrakis square tiling (a.k.a. the Union Jack lattice), which is dual to the 4.8.8 lattice [7]. 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 [8]. |
| 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. |
| Tetrahedral color code | A 3D color code defined on a colored tetrahedron cut from a suitably colored BCC lattice [9]. Qubits are placed on tetrahedra, on the triangles covering the tetrahedron faces, on the edges along the tetrahedron edges, and on the tetrahedron vertices. The code has both string-like and sheet-like logical operators [10]. |
| 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. |
| XYZ color code | A 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 \(ZZZXXX\) [11]. While such codes are equivalent to CSS color codes with the same distance, other properties like noise-bias performance can differ significantly. |
| \([[15,1,3]]\) quantum RM code | A \([[15,1,3]]\) quantum Reed-Muller code that is most easily thought of as a tetrahedral 3D color code. It can be constructed as a CSS code from the \([15,5,8]\) punctured Reed-Muller code and its even subcode, which explains its transversal \(T^\dagger\) gate [12]. |
| \([[16,6,4]]\) Tesseract color code | A (hyperbolic self-dual CSS) 4D color code defined on a tesseract, with stabilizer generators of both types supported on each cube. A \([[16,4,2,4]]\) tesseract subsystem code can be obtained from this code by using two logical qubits as gauge qubits [13]. |
| \([[17,1,5]]\) 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 \(n=17\) [14]. 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 [15]. |
| \([[2^D,D,2]]\) hypercube quantum code | Member of a family of codes defined by placing qubits on a \(D\)-dimensional hypercube, \(Z\)-stabilizers on all two-dimensional faces, and an \(X\)-stabilizer on all vertices. These codes realize gates at the \((D-1)\)-st level of the Clifford hierarchy. The measured physical bit string can be post-processed into both a logical output string and stabilizer checks, enabling end-of-circuit error detection directly from classical samples [16]. Puncturing the \([[2^D,D,2]]\) hypercube quantum code yields the \([[2^D-1,D,2]]\) punctured-hypercube family. |
| \([[2^r-1,1,3]]\) simplex code | Member of a color-code family constructed from a punctured first-order RM\((1,m=r)\) code and its even subcode for \(r \geq 3\). Each code transversally implements a diagonal gate at the \((r-1)\)st level of the Clifford hierarchy [17,18]. Each code is a color code defined on a simplex in \(r-1\) dimensions [9,19], where qubits are placed on the vertices, edges, and faces as well as on the simplex itself. |
| \([[2m,2m-2,2]]\) error-detecting code | Self-complementary and self-dual CSS code for \(m\geq 2\) with generators \(\{XX\cdots X, ZZ\cdots Z\} \) acting on all \(2m\) physical qubits. The code is constructed via the CSS construction from an SPC code and a repetition code [20; Sec. III]. This is the highest-rate distance-two code when an even number of qubits is used [21]. |
| \([[4,1,2]]\) 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 [22; Fig. 7], and equivalently as a small twist-defect surface code on a tetrahedron inscribed in a sphere [23]. 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 [24; Thm. 8]. |
| \([[6,2,2]]\) \(C_6\) 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 [25]. 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 [21; Tab. III]. Concatenations of this code with itself yield the \([[6^r,4^r,2^r]]\) level-\(r\) many-hypercube code [26]. |
| \([[7,1,3]]\) Steane code | A \([[7,1,3]]\) self-dual CSS code that is the smallest qubit CSS code to correct a single-qubit error [27]. The code is constructed using the classical binary \([7,4,3]\) Hamming code for protecting against both \(X\) and \(Z\) errors. |
| \([[8,2,2]]\) hyperbolic color code | An \([[8,2,2]]\) hyperbolic color code defined on the projective plane. It is a self-dual CSS code. |
| \([[8,3,2]]\) Smallest interesting color code | Smallest 3D color code whose physical qubits lie on vertices of a cube and which admits a (weakly) transversal \(CCZ\) gate. |
List: Union of:
- •
codes that are descendants of Color code
- •
codes that are descendants of Twist-defect color code
References
- [1]
- 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
- [2]
- M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
- [3]
- A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
- [4]
- E. B. da Silva and W. S. Soares, “Hyperbolic quantum color codes”, (2018) arXiv:1804.06382
- [5]
- N. Delfosse, “Tradeoffs for reliable quantum information storage in surface codes and color codes”, 2013 IEEE International Symposium on Information Theory 917 (2013) arXiv:1301.6588 DOI
- [6]
- C. Vuillot and N. P. Breuckmann, “Quantum Pin Codes”, IEEE Transactions on Information Theory 68, 5955 (2022) arXiv:1906.11394 DOI
- [7]
- 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
- [8]
- A. J. Landahl, J. T. Anderson, and P. R. Rice, “Fault-tolerant quantum computing with color codes”, (2011) arXiv:1108.5738
- [9]
- H. Bombin, “Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes”, (2015) arXiv:1311.0879
- [10]
- A. Kubica, M. E. Beverland, F. Brandão, J. Preskill, and K. M. Svore, “Three-Dimensional Color Code Thresholds via Statistical-Mechanical Mapping”, Physical Review Letters 120, (2018) arXiv:1708.07131 DOI
- [11]
- 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
- [12]
- D. Gottesman, Surviving as a Quantum Computer in a Classical World (2024) URL
- [13]
- B. W. Reichardt et al., “Demonstration of quantum computation and error correction with a tesseract code”, (2024) arXiv:2409.04628
- [14]
- 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
- [15]
- 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
- [16]
- D. Hangleiter, M. Kalinowski, D. Bluvstein, M. Cain, N. Maskara, X. Gao, A. Kubica, M. D. Lukin, and M. J. Gullans, “Fault-Tolerant Compiling of Classically Hard Instantaneous Quantum Polynomial Circuits on Hypercubes”, PRX Quantum 6, (2025) arXiv:2404.19005 DOI
- [17]
- B. Zeng, H. Chung, A. W. Cross, and I. L. Chuang, “Local unitary versus local Clifford equivalence of stabilizer and graph states”, Physical Review A 75, (2007) arXiv:quant-ph/0611214 DOI
- [18]
- S. X. Cui, D. Gottesman, and A. Krishna, “Diagonal gates in the Clifford hierarchy”, Physical Review A 95, (2017) arXiv:1608.06596 DOI
- [19]
- B. J. Brown, N. H. Nickerson, and D. E. Browne, “Fault-tolerant error correction with the gauge color code”, Nature Communications 7, (2016) arXiv:1503.08217 DOI
- [20]
- N. Rengaswamy, R. Calderbank, H. D. Pfister, and S. Kadhe, “Synthesis of Logical Clifford Operators via Symplectic Geometry”, 2018 IEEE International Symposium on Information Theory (ISIT) 791 (2018) arXiv:1803.06987 DOI
- [21]
- 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
- [22]
- 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
- [23]
- S. Burton, E. Durso-Sabina, and N. C. Brown, “Genons, Double Covers and Fault-tolerant Clifford Gates”, (2024) arXiv:2406.09951
- [24]
- E. M. Rains, “Quantum codes of minimum distance two”, (1997) arXiv:quant-ph/9704043
- [25]
- E. Knill, “Quantum computing with realistically noisy devices”, Nature 434, 39 (2005) arXiv:quant-ph/0410199 DOI
- [26]
- H. Goto, “High-performance fault-tolerant quantum computing with many-hypercube codes”, Science Advances 10, (2024) arXiv:2403.16054 DOI
- [27]
- 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