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 [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 \(h_1\), the 3D color code encodes \(k=3h_1\) logical qubits [1].Transversal 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 [3].Transversal \(S\) gate on color codes on general 3-manifolds corresponds to a higher-form symmetry [3].Universal transversal gates can be achieved using lattice surgery or code deformation [4,5].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 \(d=O(\log n)\), \(O(1/\log^2 n)\) with \(d=\Omega(\log^2 n)\), and \(O(1)\) with distance scaling unknown, respectively [3].Gates
Magic-state distillation protocols [6].Non-clifford gates can be implemented via code switching [6].Decoding
Decoder that maps 3D color code to three copies of the 3D surface code [7].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 [8–10]. This process can be viewed as an ungauging [11–20] of certain symmetries. This mapping can also be done via code concatenation [21]. 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 [8–10]. This process can be viewed as an ungauging [11–20] of certain symmetries. This mapping can also be done via code concatenation [21].
- 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 [3]. Transversal \(S\) gate on color codes on general 3-manifolds corresponds to a higher-form symmetry [3].
- 3D DA color code— At certain measurement rounds, the 3D DA color code realizes the instantaneous stabilizer group (ISG) of the 3D color code [22; Sec. VI.A].
- Brickwork \(XS\) stabilizer code— The brickwork \(XS\) stabilizer code can be obtained from a 3D color code [23].
- Haah cubic code (CC)— The 3D color and cubic code families both include 3D codes that do not admit string-like operators.
- 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 [24]. There is a fault-tolerant measurement-free scheme for code switching between 2D and 3D color codes [25].
- 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 [26].
Member of code lists
Primary Hierarchy
Parents
Modular-qudit lattice color codeGeneralized homological-product Lattice stabilizer CSS Stabilizer Hamiltonian-based QECC Quantum
Modular-qudit 3D color codes reduce to 3D color codes for \(q=2\).
3D color code
Children
The \([[8,3,2]]\) code is the smallest non-trivial 3D color code.
References
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Page edit log
- Victor V. Albert (2023-11-13) — most recent
- Cella Kove (2023-06-20)
Cite as:
“3D color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/3d_color