Hyperbolic tesselation code

Hyperbolic tesselation code[1]

Description

Homogeneous-space code whose codewords are superpositions of positional delta functions on the hyperbolic plane. The positions are chosen according to certain planar tessellations. Tessellation-code projections are onto a copy of an irreducible representation of the rotational symmetry group of the corresponding tessellation, known as the proper triangle group. The symmetry of the tessellation determines the easily implementable logical gate sets of this code.

Examples are the code constructed using the \(\{4,3,5\}\) tessellation realizing the binary icosahedral group \(2I\), the code constructed using the \(\{5,5,5\}\) tessellation realizing the Pauli group of a five-dimensional qudit, and the code constructed using the \(\{4,6,5\}\) tessellation realizing the single-qubit Clifford group \(2O\) [1].

Protection

Protects against sufficiently small position and momentum shifts on the hyperbolic plane. Momentum errors are represented by Laplacian eigenfunctions and are indexed by a continuous variable [1].

Gates

The logical operations of the codes are realized by rotations around certain points on the hyperbolic plane. If the hyperbolic plane is viewed as embedded in three-dimensional space, the logical operations are implemented by Gaussian operations.

Cousins

Hyperbolic sphere packing— Hyperbolic tesselation codes are quantum counterparts of hyperbolic sphere packings because they store information in quantum superpositions of points on the hyperbolic plane.
Pauli tesselation QSC— Tesselation codes are constructed using symmetry groups of tesselations of real, spherical, and hyperbolic spaces [1]. Examples include the qutrit-Pauli tesselation code, Pauli tesselation QSC, and hyperbolic tesselation code, respectively.
Qutrit-Pauli tesselation code— Tesselation codes are constructed using symmetry groups of tesselations of real, spherical, and hyperbolic spaces [1]. Examples include the qutrit-Pauli tesselation code, Pauli tesselation QSC, and hyperbolic tesselation code, respectively.

Member of code lists

Primary Hierarchy

Quantum Domain

Homogeneous-space quantum Kingdom

Homogeneous-space quantum codeQECC Quantum

Parents

Homogeneous-space quantum code

Hyperbolic tesselation codes are defined on the space of functions on the hyperbolic plane, the symmetric space \(G/H\) for \(G = SO(2,1)\) the proper Lorentz group and \(H = O(2)\).

Group-representation codeQECC Quantum

Tessellation-code projections are onto a copy of an irreducible representation of the rotational symmetry group of the corresponding tessellation, known as the proper triangle group. The symmetry of the tessellation determines the easily implementable logical gate sets of this code. The logical operations of the codes are realized by rotations around certain points on the hyperbolic plane. If the hyperbolic plane is viewed as embedded in three-dimensional space, the logical operations are implemented by Gaussian operations [1].

Hyperbolic tesselation code

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

Yixu Wang (2025-10-29) — most recent
Victor V. Albert (2025-10-29)

Cite as:

“Hyperbolic tesselation code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/tesselation

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/homogeneous/tesselation.yml.