Hyperbolic tesselation code[1]
Description
Approximate quantum code that encodes a qudit in the space of normalized functions on the hyperbolic plane and that is constructed using symmetry groups of planar tilings. Tesselation code projections are onto a copy of an irreducible representation of some group, with the code constructed using the \(\{4,3,5\}\) tesselation realizing the binary icosahedral group \(2I\) [1].Protection
Protects against sufficiently small shifts in position and momentum errors on the hyperbolic plane. Momentum errors are represented by LaPlacian eigenfunctions and are indexed by a continuous variable [1].Cousins
- Molecular code— Molecular (diatomic molecular) codes are constructed using two nested subgroups of \(SO(3)\) on the state space of a particle on \(SO(3)\) (the two-sphere) [2].
- Hyperbolic sphere packing— Hyperbolic tesselation codes are quantum analogues 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
Parents
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 \(H = O(2)\).
Tesselation code projections are onto a copy of an irreducible representation of some group, with the code constructed using the \(\{4,3,5\}\) tesselation realizing the binary icosahedral group \(2I\) [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
- [2]
- V. V. Albert, J. P. Covey, and J. Preskill, “Robust Encoding of a Qubit in a Molecule”, Physical Review X 10, (2020) arXiv:1911.00099 DOI
Page edit log
- Victor V. Albert (2025-10-29) — most recent
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.