3D surface code[1,2] 

Also known as 3D toric code, 3D cubic code.


A generalization of the Kitaev surface code defined on a 3D lattice.

3D toric code often either refers to the construction on the three-dimensional torus or is an alternative name for the general construction. The construction on surfaces with boundaries is often called the 3D surface code. There exists a rotated version of the 3D surface code, the 3D rotated surface code, akin to the (2D) rotated surface code [3]. The welded surface code [4] consists of several 3D surface codes stitched together in a way that the distance scales faster than the linear size of the system.

Related models [5,6] on lattices with certain colorability are equivalent to several copies of the 3D surface code [7].


The planar 3D surface code family on a cubic lattice of length \(L\) has parameters \([[2L(L-1)^2+L^3,1,d_X=L^2,d_Z=L]]\), while the 3D toric code has parameters \([[3L^3,3,d_X=L^2,d_Z=L]]\).

Transversal Gates

Transversal CZ and CCZ gates [8,9].


CZ gate for toric code on a Klein bottle [10].Lattice surgery [9].3D and Hybrid 2D-3D surface code computation using lattice surgery and without magic-state distillation [9].Fault-tolerant Hadamard gate using teleportation and error correction [9].


Flip decoder and its modification p-flip [11].Tensor-network decoder [12].Efficient MWPM decoder for 3D toric and 3D welded surface codes [13].Generalization of linear-time ML erasure decoder [14] to 3D surface codes [13].

Fault Tolerance

Fault-tolerant Hadamard gate using teleportation and error correction [9].

Code Capacity Threshold

Independent \(X,Z\) noise: \(12\%\) for bit-flip and \(3\%\) for phase-flip channels with MWPM decoder for 3D toric code [13], and \(17.2\%\) and \(3.3\%\) with RG decoder for 3D toric code [15].Erasure noise: \(24.8\%\) with generalization of linear-time ML erasure decoder [14] to 3D surface codes [13]. No threshold was observed for the 3D welded surface code [13].


Phenomenological noise model for the 3D toric code: \(2.90(2)\%\) under BP-OSD decoder [16], \(7.1\%\) under improved BP-OSD [17], \(7.3\%\) under RG [15], and \(2.6\%\) under flip decoder [11]. For 3D surface code: \(3.08(4)\%\) under flip decoder [16].



  • Chamon model code — The Chamon and 3D surface codes can both be built out of a hypergraph product of three repetition codes; see [19; Sec. 3.4].
  • Rotated surface code — There exists a rotated version of the 3D surface code, akin to the (2D) rotated surface code [3].
  • Self-correcting quantum code — The 3D welded surface code is partially self-correcting with a power-law energy barrier [4]. The 3D toric code is a classical self-correcting memory, whose protected bit admits a membrane-like logical operator [20], but it is not a quantum self-correcting memory [21].
  • 3D color code — The 3D color code is equivalent to multiple decoupled copies of the 3D surface code via a local constant-depth Clifford circuit [8,22,23]. This mapping can also be done via code concatenation [9].
  • 3D fermionic surface code — The 3D (fermionic) surface code is a CSS (non-CSS) code which realizes a \(\mathbb{Z}_2\) gauge theory in 3D (with an emergent fermion).
  • 3D subsystem surface code


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K. P. Michnicki, “3D Topological Quantum Memory with a Power-Law Energy Barrier”, Physical Review Letters 113, (2014) arXiv:1406.4227 DOI
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Zoo Code ID: 3d_surface

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“3D surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/3d_surface
@incollection{eczoo_3d_surface, title={3D surface code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/3d_surface} }
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“3D surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/3d_surface

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/topological/surface/higher_d/3d_surface.yml.