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3D color code [1]

Description

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].

For a closed 3-manifold, the code encodes \(k=3h_1\) logical qubits, where \(h_1\) is the first Betti number over \(\mathbb{Z}_2\) [1]. Logical operators can be represented by colored strings and colored membranes. Excitations consist of point-like color charges at cell defects and loop-like color fluxes at face defects; winding a \(p\)-charge around a \(pq\)-flux produces a \(-1\) phase [1].

There are 101 different types of boundaries for any uniform tiling [2]; this was shown for the great rhombated cubic honeycomb (a.k.a. cantitruncated cubic honeycomb) uniform tiling, but is valid for general uniform tilings.

Protection

On a closed 3-manifold with first Betti number \(h_1\) over \(\mathbb{Z}_2\), the 3D color code encodes \(k=3h_1\) logical qubits [1]. In tetrahedral and cubic 3D color codes, \(Z\)-type logical operators are strings along a border where two boundaries meet, and \(X\)-type logical operators are membranes covering a boundary [3]. The minimum weight of an \(X\)-type logical operator therefore scales as the square of the code distance [3].

Transversal and Permutation-Based Gates

Transversal action of \(T\) gates on color codes on general 3-manifolds realizes a \(CCZ\) gate on three logical qubits and is related to a topological invariant that is called the triple intersection number; this gate is related to the fact that this code admits a cup product structure [4].Transversal \(S\) gate on color codes on general 3-manifolds corresponds to a higher-form symmetry [4].Families of 3D color codes on quasi-hyperbolic, fibre-bundle, and Torelli mapping-torus 3-manifolds support collective logical \(CCZ\) gates via transversal \(T\) and individually addressable, parallelizable logical \(CZ\) gates via transversal \(S\) on codimension-1 submanifolds. Their rate-distance scalings are \(O(1/\log n)\) with \(d=O(\log n)\), \(O(1/\log^{1/2} n)\) with \(d=\Omega(\log^{1/2} n)\), and \(O(1)\), respectively [4].

Gates

Universal fault-tolerant computation using transversal gates and lattice surgery [4].Magic-state distillation protocols [5].Non-Clifford gates can be implemented via code switching [5].

Decoding

Decoder that projects the 3D color code onto a collection of 3D surface codes, decoding bit-flip errors on one surface code per color and phase-flip errors on one surface code per pair of colors [6].3D concatenated MWPM decoder for codes with boundaries [3], a 3D extension of the concatenated MWPM decoder for 2D color codes [7]. It decodes \(X\) errors from face syndromes merged into cell syndromes, which reduces the effective \(X\) distance to the \(Z\) distance [3].

Code Capacity Threshold

Bit- and phase-flip noise on the bcc lattice: \(0.77\%\) for point-like and \(13.1\%\) for loop-like excitations under restriction decoder [8].Bit- and phase-flip noise on the \([[5d^3-12d^2+16,3,d]]\) cubic color code family [9]: \(1.55(6)\%\), with pseudo-threshold \(0.60(6)\%\) and logical error rate scaling as \(p^{d/2}\), under the 3D concatenated MWPM decoder [3].

Notes

See the qCodePlot3D Python package for visualizations of 3D color codes and their decoding paths [3].

Cousins

  • 3D surface code— On closed 3-manifolds, the 3D color code is equivalent to multiple decoupled copies of the 3D surface code via a local constant-depth Clifford circuit [10–12]. This process can be viewed as an ungauging [13–22] of certain symmetries. This mapping can also be done via code concatenation [23]. In contrast to the 3D surface/toric code, the original colex Hamiltonian can be viewed as both a string-net condensate and a membrane-net condensate [1].
  • Concatenated qubit code— On closed 3-manifolds, the 3D color code is equivalent to multiple decoupled copies of the 3D surface code via a local constant-depth Clifford circuit [10–12]. This process can be viewed as an ungauging [13–22] of certain symmetries. This mapping can also be done via code concatenation [23].
  • XS stabilizer code— The 3D color code on a particular lattice admits XS stabilizers; see talk by M. Kesselring at the 2020 FTQC conference.
  • Symmetry-protected topological (SPT) code— Transversal action of \(T\) gates on color codes on general 3-manifolds realizes a \(CCZ\) gate on three logical qubits and is related to a topological invariant that is called the triple intersection number [4]. Transversal \(S\) gate on color codes on general 3-manifolds corresponds to a higher-form symmetry [4].
  • 3D DA color code— At certain measurement rounds, the 3D DA color code realizes the instantaneous stabilizer group (ISG) of the 3D color code [24; Sec. VI.A].
  • Brickwork \(XS\) stabilizer code— The brickwork \(XS\) stabilizer code can be obtained from a 3D color code [25].
  • Haah cubic code (CC)— The 3D color and cubic code families both include 3D codes that do not admit string-like operators.
  • Tricycle code— Both 3D color codes and tricycle codes admit constant-depth \(CCZ\) circuits stemming from cup products on three-dimensional chain complexes [26]. Tricycle codes attain much higher rates at comparable block lengths [26].
  • 2D 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 [27]. There is a fault-tolerant measurement-free scheme for code switching between 2D and 3D color codes [28].
  • 3D XYZ color code— The 3D XYZ color code is defined on the same lattices as the 3D color code, with \(X\)-, \(Y\)-, or \(Z\)-type face generators fixed by color labels. On a closed 3-manifold, it encodes \(b_2\) logical qubits, while the 3D color code encodes \(3b_2\) [29; Sec. 3.4].
  • 3D subsystem color code— On a fixed 3D lattice, the 3D subsystem color code is gauge-related to the 3D color code; switching between the \((1,1)\) and \((1,2)\) members yields transversal \(CNOT\), \(H\), and \(R_3\) gates [30].

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
3D lattice stabilizer codeLattice stabilizer Stabilizer Hamiltonian-based QECC Quantum
Modular-qudit lattice color codeCSS Generalized homological-product Lattice stabilizer Stabilizer Hamiltonian-based QECC Quantum
Modular-qudit 3D color codes reduce to 3D color codes for \(q=2\).
Abelian topological codeTopological Hamiltonian-based QECC Quantum
3D color code
Children
\([[8,3,2]]\) Smallest interesting color code
The \([[8,3,2]]\) code is the smallest non-trivial 3D color code.
Cubic honeycomb color code
Tetrahedral color code

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Zoo Code ID: 3d_color

Cite as:
“3D color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/3d_color, arXiv:2606.11484
BibTeX:
@incollection{eczoo_3d_color,
title={3D 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/3d_color}
}
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“3D color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/3d_color, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/topological/color/3d_color/3d_color.yml.