Hyperbolic tessellation code[1]
Description
Homogeneous-space code whose codewords are superpositions of positional delta functions on the hyperbolic plane. The positions are chosen according to regular triangle tessellations, and the code projector picks out an irreducible representation of the corresponding proper triangle group.
Examples in Ref. [1] are the \(\{5,5,5\}\) tessellation realizing the Pauli group of a five-dimensional qudit, the \(\{6,4,8\}\) tessellation realizing the single-qubit Clifford group, and the \(\{4,3,5\}\) tessellation realizing the binary icosahedral group \(2I\).
Protection
Protects against sufficiently small position and momentum shifts on the hyperbolic plane. In the three explicit examples of Ref. [1], position-translation errors are correctable up to resolutions \(d_x \approx 1.6169\), \(0.6605\), and \(0.5011\), respectively. Momentum errors are represented by hyperbolic Laplacian eigenfunctions; the \(\{5,5,5\}\) example corrects all modes with angular index \(n<5\), and more generally the compact quotient \(\mathbb{H}^2/\Gamma\) provides a nonzero Laplacian gap that acts as a momentum-error distance scale.Gates
Logical operations are realized by rotations around selected vertices of the hyperbolic tessellation. In the \(\{6,4,8\}\) example, \(S\) and \(U\) are implemented by \(\pi/4\) and \(\pi/3\) rotations, while the \(\{4,3,5\}\) example realizes binary-icosahedral non-Clifford gates by \(2\pi/3\) and \(2\pi/5\) rotations [1].Cousins
- Hyperbolic sphere packing— Hyperbolic tessellation codes are quantum counterparts of hyperbolic sphere packings because they store information in quantum superpositions of points on the hyperbolic plane.
- Pauli tessellation QSC— The tessellation-code framework spans spherical, Euclidean, and hyperbolic geometries; the Pauli tessellation QSC is the spherical member [1].
- Qutrit-Pauli tessellation code— The qutrit-Pauli tessellation code is the Euclidean \(\{3,3,3\}\) member of the same curvature-dependent tessellation-code framework [1].
- Concatenated GKP code— The qubit-Pauli tessellation GKP code [1] is the Euclidean \(\{2,4,4\}\) member of the curvature-dependent tessellation-code framework. It is a two-mode code in which each Cartesian direction is a single-mode qubit GKP code, making the full code a 2-to-1 concatenated qubit encoding. The logical single-qubit Pauli group is implemented geometrically by one \(\pi\) rotation and two \(\pi/2\) rotations on the Euclidean tessellation [1].
Member of code lists
Primary Hierarchy
References
- [1]
- Y. Wang, Y. Xu, and Z.-W. Liu, “Tessellation Codes: Encoded Quantum Gates by Geometric Rotation”, Physical Review Letters 135, (2025) arXiv:2410.18713 DOI
Page edit log
- Victor V. Albert (2026-04-22) — most recent
- Yixu Wang (2025-10-29)
- Victor V. Albert (2025-10-29)
Cite as:
“Hyperbolic tessellation code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/tesselation
Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/homogeneous/tesselation.yml.