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2D color code [1,2]

Description

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.

Most translation-invariant color codes are defined on trivalent planar graphs with three-colorable faces. The three admissible uniform tilings are the 6.6.6 (honeycomb) tiling, the 4.8.8 (square octagon) tiling, and the 4.6.12 tiling [3; Fig. 1]. The smallest distance-four codes on tori tiled in these three ways are the \([[18,4,4]]\) code, the \([[16,4,4]]\) code, and a \([[36,4,4]]\) code, respectively. Non-uniform tilings include the [4.6.8, 6.8.8] and [4.6.8, 4.8.12] tilings [4]. More general admissible tilings can be obtained via a fattening procedure [2]; see also a construction based on the more general quantum pin codes [5].

Logical dimension is determined by the genus of the underlying surface (for closed surfaces) and the types of boundaries (for open surfaces). There are six basic boundary types: three color boundaries corresponding to the three face colors [6] and three Pauli boundaries corresponding to the three Pauli labels [7].

String operators are defined on paths along edges of the qubit lattice. These paths can have branching points. Each path has two string operators, one corresponding to the \(X\) basis and one corresponding to the \(Z\) basis. In correspondence with the coloring of the lattice faces, string operators also come in three colors. A string of one color must end in a boundary of that same color.

Rate

For general 2D manifolds, \(kd^2 \leq c(\log k)^2 n\) for some constant \(c\) in what can be thought of as an extension of the BPT bound to codes on hyperbolic geometries [8], meaning that color codes with finite rate can only achieve an asymptotic minimum distance that is logarithmic in \(n\).

Transversal and Permutation-Based Gates

CNOT gate because the code is CSS.Hadamard gates for any qubit geometry which yields a self-dual CSS code.Certain triangular 2D color codes on suitably chosen lattices admit transversal implementations of the full Clifford group, including \(H\), \(S^\dagger\), and \(CNOT\), without selective addressing [1].

Gates

Magic-state distillation protocols [9].Non-Clifford gates can be implemented via code switching [9].

Decoding

Projection decoder of \(O(n^4)\) complexity [10], modified to account for syndrome errors [11].Exact minimum-weight decoding is \(NP\)-hard for 2D color codes, including for solely Pauli-\(Z\) noise [12,13].Chromöbius, an open-source implementation of the Möbius decoder, works for many 2D color codes [14].Concatenated MWPM decoder [15].Syndrome extraction circuits based on superdense coding and a middle-out strategy [14].

Cousins

  • Kitaev surface code— On closed surfaces, the 2D color code is equivalent to two decoupled copies of the 2D toric/surface code via a local constant-depth Clifford circuit [16–18] and has the same topological entanglement entropy [19]. For triangular patches with three differently colored boundaries, it is instead equivalent to a folded surface/toric code with two smooth and two rough boundaries [17]. The conversion process can be viewed as an ungauging [20–29] of certain symmetries. Conversely, the 2D color code can condense to form the 2D surface code in nine different ways, i.e., by adding two-body hopping terms along one of its three triangular directions to the stabilizer group and then taking the center of the resulting nonabelian group [30]. Both the surface and 2D color codes can be constructed from two distinct types of lattices, namely, 4-valent and 3-valent 3-colorable lattices, respectively [31].
  • 3D color code— Gauge fixing can be used to code switch between 2D and 3D color codes, thereby yielding fault-tolerant computation with constant time overhead using only local quantum operations [32]. There is a fault-tolerant measurement-free scheme for code switching between 2D and 3D color codes [33].
  • Linear binary code— As CSS codes, variants of the 2D color code are constructed out of self-dual classical codes on cubic planar graphs [34].
  • Hamiltonian-based code— 2D color code Hamiltonians can be simulated, with the help of perturbation theory, by two-dimensional weight-two (two-body) Hamiltonians with non-commuting terms [35].
  • 2D DA color code— At certain measurement rounds, the 2D DA color code realizes the instantaneous stabilizer group (ISG) of the 2D color code [36; Sec. III.A].
  • Floquet color code— The parent topological phase of the Floquet color code is the \(\mathbb{Z}_2\times\mathbb{Z}_2\) 2D color-code phase.
  • Ruby Floquet code— Each ISG of the color-code schedule is FDLQC-equivalent to the 2D color code, and a parent stabilizer code is FDLQC-equivalent to two copies of the 2D color code [37].
  • Floquet 3D surface code— A parent stabilizer code for the Floquet 3D surface code is FDQC-equivalent to a 3-foliated stack of 2D color codes [37].
  • X-cube Floquet code— A parent stabilizer code for the rewinding X-cube Floquet code is FDQC-equivalent to a 3-foliated stack of 2D color codes [37].
  • Majorana color code— The original Majorana color code is a fermionic analogue of a 2D color code in which one Majorana face operator doubles to matching \(X\)- and \(Z\)-type face checks, but the underlying cylinder graph need only be locally \(3\)-colorable and can support odd boundary logical operators [38]. Later realizations stack Majorana surface-code layers and replace stacked building blocks with small Majorana fermion codes [4,39–41].
  • Derby-Klassen (DK) code— The DK code on several tilings resembles the 2D color code with some vertex qubits removed [42].
  • Barbell quantum code— Barbell syndrome extraction adapts the superdense syndrome-extraction circuits originally developed to implement color codes on a square grid [14].
  • Directional tile code— The boundary-dominated routing overhead of directional tile codes is analogous to the routing overhead established for color-code circuits built from CXSWAP gates [43].
  • 2D subsystem color code— Gauge fixing relates 2D subsystem color codes to 2D color codes on the same lattice [6,44]; the original Union-Jack member is reproduced by the square-octagon-lattice construction of [45].
  • Three-fermion (3F) subsystem code— The 2D color code is equivalent to two decoupled copies of the 3F code in the sense that the same anyon theory describes the low-energy excitations of both codes [46][7; Appx. B].

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Qubit stabilizer codeQubit Stabilizer Hamiltonian-based QECC Quantum
Qubit QLDPC codeStabilizer Hamiltonian-based QECC Quantum
Color codeQubit CSS Stabilizer Hamiltonian-based QECC Quantum
Parents
Color code
Generalized 2D color codeTopological Hamiltonian-based QECC Quantum
The generalized color code for \(G=\mathbb{Z}_2\) reduces to the 2D color code.
Twist-defect color codeQLDPC Qubit Lattice stabilizer Stabilizer Topological Hamiltonian-based QECC Quantum
Twist-defect color codes reduce to 2D color codes when there are no defects. See Ref. [47] for an alternative non-CSS extension of 2D color codes.
Modular-qudit lattice color codeCSS Generalized homological-product Lattice stabilizer Stabilizer Hamiltonian-based QECC Quantum
Modular-qudit 2D color codes reduce to 2D color codes for \(q=2\).
Galois-qudit color codeCSS Generalized homological-product Lattice stabilizer Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum
Galois-qudit 2D color codes reduce to 2D color codes for \(q=2\).
Abelian quantum-double stabilizer codeLattice stabilizer Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum
When treated as ground states of the code Hamiltonian, states of the color code on a torus geometry realize \(\mathbb{Z}_2\times\mathbb{Z}_2\) topological order [35], equivalent to the phase realized by two copies of the toric code (i.e., the surface code on a torus) via a local constant-depth Clifford circuit [17]. This process can be viewed as an ungauging [20–29] of certain symmetries.
2D color code
Children
\([[6,2,2]]\) \(C_6\) code
The \(C_6\) code is a color code on a ladder with three rungs and periodic boundary conditions, (a.k.a. a triangular prism with no top and bottom faces) [48]. Purely \(Z\)- or \(X\)-type stabilizers lie on the three square faces of the ladder.
\([[8,2,2]]\) projective-plane color code
The \([[8,2,2]]\) projective-plane color code comes from a tiling of the projective plane with octahedral symmetry, whose checks are one octagon and four squares [5; Sec. V.A].
Truncated trihexagonal (4.6.12) color code
Square-octagon (4.8.8) color code
Honeycomb (6.6.6) color code

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Zoo Code ID: 2d_color

Cite as:
“2D color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/2d_color, arXiv:2606.11484
BibTeX:
@incollection{eczoo_2d_color,
title={2D color code},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/2d_color}
}
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“2D color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/2d_color, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/topological/color/2d_color/2d_color.yml.