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\((1,3)\) 4D toric code[1]

Description

A generalization of the Kitaev surface code defined on a 4D lattice. The code is called a \((1,3)\) toric code because it admits 1D \(Z\)-type and 3D \(X\)-type logical operators. In the hypercubic lattice version, qubits are placed on edges, each \(Z\)-type stabilizer generator is supported on cubes on the boundary of a hypercube, and \(X\)-type stabilizers are placed on the edges neighboring every vertex [1].

Transversal and Permutation-Based Gates

Logical \(CCCZ\) gate on a hyper-diamond lattice [1].

Cousins

  • \(D_4\) hyper-diamond lattice— The \((1,3)\) 4D toric code on a hyper-diamond lattice admits a transversal logical \(CCCZ\) gate [1].
  • Higher-dimensional homological product code— The 4D \((1,3)\) planar (toric) code on a hypercubic lattice can be obtained from a particular choice of chain complex from a hypergraph product of four repetition codes [2].
  • Repetition code— The 4D \((1,3)\) planar (toric) code on a hypercubic lattice can be obtained from a particular choice of chain complex from a hypergraph product of four repetition codes [2].

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum
Coherent-parity-check (CPC) code
Qubit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum
Generalized homological-product qubit CSS codeQLDPC Qubit Generalized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
Homological code
Parents
Homological code
The \((1,3)\) 4D toric code realizes 4D \(\mathbb{Z}_2\) gauge theory with 1D \(Z\)-type and 3D \(X\)-type logical operators.
4D lattice stabilizer codeLattice stabilizer Stabilizer Hamiltonian-based QECC Quantum
Abelian topological codeTopological Hamiltonian-based QECC Quantum
The \((1,3)\) 4D toric code realizes 4D \(\mathbb{Z}_2\) gauge theory with 1D \(Z\)-type and 3D \(X\)-type logical operators.
Dijkgraaf-Witten gauge theory codeTopological Hamiltonian-based QECC Quantum
An untwisted Dijkgraaf-Witten theory in 4D for the group \(G=\mathbb{Z}_2\) is a \((1,3)\) 4D toric code.
\((1,3)\) 4D toric code

References

[1]
T. Jochym-O’Connor and T. J. Yoder, “Four-dimensional toric code with non-Clifford transversal gates”, Physical Review Research 3, (2021) arXiv:2010.02238 DOI
[2]
W. Zeng and L. P. Pryadko, “Higher-Dimensional Quantum Hypergraph-Product Codes with Finite Rates”, Physical Review Letters 122, (2019) arXiv:1810.01519 DOI
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Zoo Code ID: 4d_13_surface

Cite as:
“\((1,3)\) 4D toric code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/4d_13_surface, arXiv:2606.11484
BibTeX:
@incollection{eczoo_4d_13_surface,
title={\((1,3)\) 4D toric 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/4d_13_surface}
}
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Permanent link:
https://errorcorrectionzoo.org/c/4d_13_surface

Cite as:

“\((1,3)\) 4D toric code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/4d_13_surface, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/topological/surface/higher_d/4d_13_surface.yml.