Also known as Triangular color code.
Description
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.
Most translation-invariant color codes are defined in 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]. More general admissible tilings can be obtained via a fattening procedure [2]; see also a construction based on the more general quantum pin codes [4].
Logical dimension is determined by the genus of the underlying surface (for closed surfaces) and the types of boundaries (for open surfaces). There are three types of boundaries corresponding to the three colors of the faces [5,6].
String operators are defined on the paths along the edges of the qubit. 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 [7], meaning that color codes with finite rate can only achieve an asymptotic minimum distance that is logarithmic in \(n\).
Transversal Gates
CNOT gate because the code is CSS.Hadamard gates for any qubit geometry which yields a self-dual CSS code.Transversal \(S\) gate [1,3].
Gates
Magic-state distillation protocols [8].Non-clifford gates can be implemented via code switching [8].
Decoding
Projection decoder of \(O(n^4)\) complexity [9], modified to account for syndrome errors [10].Concatenated MPWM decoder [11].Syndrome extraction circuits based on superdense coding and a middle-out strategy [12].
Parents
- Color code
- Generalized 2D color code — The generalized color code for \(G=\mathbb{Z}_2\) reduces to the 2D color code.
- Twist-defect color code — Twist-defect color codes reduce to 2D color codes when there are no defects. See Ref. [13] for an alternative non-CSS extension of 2D color codes.
- Modular-qudit color code — Modular-qudit 2D color codes reduce to 2D color codes for \(q=2\).
- Galois-qudit color code — Galois-qudit 2D color codes reduce to 2D color codes for \(q=2\).
- Abelian quantum-double stabilizer code — 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 [14], equivalent to the phase realized by two copies of the surface code via a local constant-depth Clifford circuit [15]. This process can be viewed as an ungauging [16–18,18] of certain symmetries.
Children
- \([[4,2,2]]\) Four-qubit code — The \([[4,2,2]]\) code can be interpreted as a 2D color code on a square of the 4.8.8 or 4.6.12 tilings, or on a trapezoidal patch that makes up two-thirds of a hexagon of the 6.6.6 tiling [19,20]. Concatenating the \([[4,2,2]]\) code with the surface code is equivalent to removing stabilizer generators from the 4.8.8 color code [21]. Concatenating the \([[4,2,2]]\) code with two copies of the surface code yields the 4.6.12 color code [21]. A small 6.6.6 color code is a \([[4,1,2]]\) subcode with three weight-three stabilizer generators [6; Fig. 7]; this code is equivalent to a twist-defect surface code on a tetrahedron inscribed in a sphere [22] via a single-qubit Clifford circuit.
- Truncated trihexagonal (4.6.12) color code
- Square-octagon (4.8.8) color code
- Union-Jack color code
- Honeycomb (6.6.6) color code
Cousins
- Kitaev surface code — The 2D color code is equivalent to multiple decoupled copies of the 2D surface code via a local constant-depth Clifford circuit [15,23,24]. This process can be viewed as an ungauging [16–18,18] 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 [19]. 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 [25].
- 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 [26]. There is a fault-tolerant measurement-free scheme for code switching between 2D and 3D color codes [27].
- Linear binary code — As CSS codes, variants of the 2D color code are constructed out of self-dual classical codes on cubic planar graphs [28].
- 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 [14].
- 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.
- Majorana color code — The Majorana color code is a Majorana stabilizer analogue of the 2D color code.
- Derby-Klassen (DK) code — The DK code on several tilings resembles the 2D color code with some vertex qubits removed [29].
- 2D subsystem color code
- 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 [30][6; Appx. B].
References
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Page edit log
- Victor V. Albert (2023-11-13) — most recent
- Cella Kove (2023-06-20)
Cite as:
“2D color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/2d_color