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arXiv:2410.18713 · Physical Review Letters

Tessellation Codes: Encoded Quantum Gates by Geometric Rotation

Yixu Wang, Yijia Xu, Zi-Wen Liu

Published Aug 13, 2025• 1 citation
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Abstract

We utilize the symmetry groups of regular tessellations on two-dimensional surfaces of different constant curvatures, including spheres, Euclidean planes, and hyperbolic planes to encode a qubit or qudit into the physical degrees of freedom on these surfaces, which we call tessellation codes. We show that tessellation codes exhibit decent error correction properties by analysis via geometric considerations and the representation theory of the isometry groups on the corresponding surfaces. Interestingly, we demonstrate how this formalism enables the implementation of certain logical operations through geometric rotations of surfaces in real space, opening a new approach to logical quantum computation. We provide a variety of concrete constructions of such codes associated with different tessellations, which give rise to different logical groups. This formalism sheds a new light on quantum code and logical operation construction.

DOI
10.1103/tljb-f7tt

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We utilize the symmetry groups of regular tessellations on two-dimensional surfaces of different constant curvatures, including spheres, Euclidean planes, and hyperbolic planes to encode a qubit or qudit into the physical degrees of freedom on these surfaces, which we call tessellation codes. We show that tessellation codes exhibit decent error correction properties by analysis via geometric considerations and the representation theory of the isometry groups on the corresponding surfaces. Interestingly, we demonstrate how this formalism enables the implementation of certain logical operations through geometric rotations of surfaces in real space, opening a new approach to logical quantum computation. We provide a variety of concrete constructions of such codes associated with different tessellations, which give rise to different logical groups. This formalism sheds a new light on quantum code and logical operation construction.
We introduce group surface codes, which are a natural generalization of the Z2\mathbb{Z}_2Z2​ surface code, and equivalent to quantum double models of finite groups with specific boundary conditions. We show that group surface codes can be leveraged to perform non-Clifford gates in Z2\mathbb{Z}_2Z2​ surface codes, thus enabling universal computation along with established means of performing logical Clifford gates. Moreover, for suitably chosen groups, we demonstrate that arbitrary reversible classical gates can be implemented transversally in the group surface code. We present the logical operations in terms of a set of elementary logical operations, which include transversal logical gates, a means of transferring encoded information into and out of group surface codes, and preparation and readout. By composing these elementary operations, we implement a wide variety of logical gates and provide a unified perspective on recent constructions in the literature for sliding group surface codes and preparing magic states. We furthermore use tensor networks inspired by ZX-calculus to construct spacetime implementations of the elementary operations. This spacetime perspective also allows us to establish explicit correspondences with topological gauge theories. Our work extends recent efforts in performing universal quantum computation in topological orders without the braiding of anyons, and shows how certain group surface codes allow us to bypass the restrictions set by the Bravyi-König theorem, which limits the computational power of topological Pauli stabilizer models.
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