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

\((1,3)\) 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.

Transversal Gates

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

Cousin

\(D_4\) hyper-diamond lattice— The \((1,3)\) 4D toric code on a hyper-diamond lattice admits a transversal logical \(CCCZ\) gate [1].

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 Qubit Hamiltonian-based QECC Quantum

Generalized homological-product qubit CSS codeGeneralized homological-product QLDPC 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.

Lattice stabilizer codeQLDPC 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.

\((1,3)\) 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

Page edit log

Victor V. Albert (2024-07-10) — most recent
Nathanan Tantivasadakarn (2024-07-10)

Cite as:

“\((1,3)\) 4D toric code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/4d_13_surface

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