3D surface code[1,2] 

Also known as 3D toric code, 3D cubic code, Bosonic-charge bosonic-loop (BcBl) surface code.

Description

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 planar 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].

Protection

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 \([[3L^3,3,d_X=L^2,d_Z=L]]\).

Stability against Hamiltonian perturbations was determined using a tensor-network representation [8]. The phase diagram of the perturbed tensor network maps to that of a 3D Ising gauge theory.

Transversal Gates

Transversal CZ and CCZ gates [9,10].

Gates

CZ gate for toric code on a Klein bottle [11].Lattice surgery [10].3D and Hybrid 2D-3D surface code computation using lattice surgery and without magic-state distillation [10].Fault-tolerant Hadamard gate using teleportation and error correction [10].

Decoding

Flip decoder and its modification p-flip [12].Tensor-network decoder [13].Efficient MWPM decoder for 3D toric and 3D welded surface codes [14].Generalization of linear-time ML erasure decoder [15] to 3D surface codes [14].

Fault Tolerance

Fault-tolerant Hadamard gate using teleportation and error correction [10].

Code Capacity Threshold

Independent \(X,Z\) noise: \(12\%\) for bit-flip and \(3\%\) for phase-flip channels with MWPM decoder for 3D toric code [14], and \(17.2\%\) and \(3.3\%\) with RG decoder for 3D toric code [16].Erasure noise: \(24.8\%\) with generalization of linear-time ML erasure decoder [15] to 3D surface codes [14]. No threshold was observed for the 3D welded surface code [14].

Threshold

Phenomenological noise model for the 3D toric code: \(2.90(2)\%\) under BP-OSD decoder [17], \(7.1\%\) under improved BP-OSD [18], \(7.3\%\) under RG [16], and \(2.6\%\) under flip decoder [12]. For 3D surface code: \(3.08(4)\%\) under flip decoder [17].

Parents

Cousins

  • Chamon model code — The Chamon and 3D surface codes can both be built out of a hypergraph product of three repetition codes; see [21; Sec. 3.4].
  • Rotated surface code — There exists a rotated version of the 3D surface code, akin to the (2D) rotated surface code [3].
  • Hamiltonian-based code — Stability of the 3D surface code against Hamiltonian perturbations was determined using a tensor-network representation [8]. The phase diagram of the perturbed tensor network maps to that of a 3D Ising gauge theory.
  • 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 [22], but it is not a quantum self-correcting memory [23].
  • \([[8,3,2]]\) CSS code — The \([[8,3,2]]\) code can be concatenated with a 3D surface code to yield a \([[O(d^3),3,2d]]\) code family that admits a transversal implementation of the logical \(CCZ\) gate [24].
  • 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 [9,25,26]. This process can be viewed as an ungauging [27–29,29] of certain symmetries. This mapping can also be done via code concatenation [10].
  • 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). Two copies of the 3D fermionic surface code are equivalent to a copy of the 3D surface code and a copy of the 3D fermionic surface code via anyon relabeling: the two incoming fermions, \(\f_1\) and \(f_2\), can be re-organized into a boson \(f_1 f_2\) and fermion \(f_2\).
  • Fractal surface code — Fractal surface codes are obtained by removing qubits from the 3D surface code on a cubic lattice.
  • 3D subsystem surface code

References

[1]
E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
[2]
A. Hamma, P. Zanardi, and X.-G. Wen, “String and membrane condensation on three-dimensional lattices”, Physical Review B 72, (2005) arXiv:cond-mat/0411752 DOI
[3]
E. Huang, A. Pesah, C. T. Chubb, M. Vasmer, and A. Dua, “Tailoring Three-Dimensional Topological Codes for Biased Noise”, PRX Quantum 4, (2023) arXiv:2211.02116 DOI
[4]
K. P. Michnicki, “3D Topological Quantum Memory with a Power-Law Energy Barrier”, Physical Review Letters 113, (2014) arXiv:1406.4227 DOI
[5]
H. Bombin and M. A. Martin-Delgado, “Exact topological quantum order inD=3and beyond: Branyons and brane-net condensates”, Physical Review B 75, (2007) arXiv:cond-mat/0607736 DOI
[6]
I. H. Kim, “Local non-CSS quantum error correcting code on a three-dimensional lattice”, (2013) arXiv:1012.0859
[7]
A. Dua, I. H. Kim, M. Cheng, and D. J. Williamson, “Sorting topological stabilizer models in three dimensions”, Physical Review B 100, (2019) arXiv:1908.08049 DOI
[8]
D. J. Williamson, C. Delcamp, F. Verstraete, and N. Schuch, “On the stability of topological order in tensor network states”, Physical Review B 104, (2021) arXiv:2012.15346 DOI
[9]
A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
[10]
M. Vasmer and D. E. Browne, “Three-dimensional surface codes: Transversal gates and fault-tolerant architectures”, Physical Review A 100, (2019) arXiv:1801.04255 DOI
[11]
M. Barkeshli, Y.-A. Chen, P.-S. Hsin, and R. Kobayashi, “Higher-group symmetry in finite gauge theory and stabilizer codes”, SciPost Physics 16, (2024) arXiv:2211.11764 DOI
[12]
T. R. Scruby and K. Nemoto, “Local Probabilistic Decoding of a Quantum Code”, Quantum 7, 1093 (2023) arXiv:2212.06985 DOI
[13]
C. Piveteau, C. T. Chubb, and J. M. Renes, “Tensor-Network Decoding Beyond 2D”, PRX Quantum 5, (2024) arXiv:2310.10722 DOI
[14]
A. Kulkarni and P. K. Sarvepalli, “Decoding the three-dimensional toric codes and welded codes on cubic lattices”, Physical Review A 100, (2019) arXiv:1808.03092 DOI
[15]
N. Delfosse and G. Zémor, “Linear-time maximum likelihood decoding of surface codes over the quantum erasure channel”, Physical Review Research 2, (2020) arXiv:1703.01517 DOI
[16]
K. Duivenvoorden, N. P. Breuckmann, and B. M. Terhal, “Renormalization Group Decoder for a Four-Dimensional Toric Code”, IEEE Transactions on Information Theory 65, 2545 (2019) arXiv:1708.09286 DOI
[17]
A. O. Quintavalle, M. Vasmer, J. Roffe, and E. T. Campbell, “Single-Shot Error Correction of Three-Dimensional Homological Product Codes”, PRX Quantum 2, (2021) arXiv:2009.11790 DOI
[18]
O. Higgott and N. P. Breuckmann, “Improved Single-Shot Decoding of Higher-Dimensional Hypergraph-Product Codes”, PRX Quantum 4, (2023) arXiv:2206.03122 DOI
[19]
J. Haah, “A degeneracy bound for homogeneous topological order”, SciPost Physics 10, (2021) arXiv:2009.13551 DOI
[20]
L. Berent, T. Hillmann, J. Eisert, R. Wille, and J. Roffe, “Analog Information Decoding of Bosonic Quantum Low-Density Parity-Check Codes”, PRX Quantum 5, (2024) arXiv:2311.01328 DOI
[21]
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[22]
O. Landon-Cardinal, B. Yoshida, D. Poulin, and J. Preskill, “Perturbative instability of quantum memory based on effective long-range interactions”, Physical Review A 91, (2015) arXiv:1501.04112 DOI
[23]
Y. Li, C. W. von Keyserlingk, G. Zhu, and T. Jochym-O’Connor, “Phase diagram of the three-dimensional subsystem toric code”, Physical Review Research 6, (2024) arXiv:2305.06389 DOI
[24]
D. Hangleiter, M. Kalinowski, D. Bluvstein, M. Cain, N. Maskara, X. Gao, A. Kubica, M. D. Lukin, and M. J. Gullans, “Fault-tolerant compiling of classically hard IQP circuits on hypercubes”, (2024) arXiv:2404.19005
[25]
B. Yoshida, “Classification of quantum phases and topology of logical operators in an exactly solved model of quantum codes”, Annals of Physics 326, 15 (2011) arXiv:1007.4601 DOI
[26]
A. B. Aloshious, A. N. Bhagoji, and P. K. Sarvepalli, “On the Local Equivalence of 2D Color Codes and Surface Codes with Applications”, (2018) arXiv:1804.00866
[27]
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[28]
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[29]
W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 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
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@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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