Tessellation codes: encoded quantum gates by geometric rotation

TL;DR

It is demonstrated how the symmetry groups of regular tessellations on two-dimensional surfaces of different constant curvatures enables the implementation of certain logical operations through geometric rotations of surfaces in real space, opening a new approach to logical quantum computation.

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.

Figures (8)

• Figure 1: The $\{2,4,4\}$ tessellation on the plane is used to illustrate the action of the generators of the triangle and proper triangle groups. Generators of other $\{p,q,r\}$ groups act similarly geometrically. The dashed double-arrow lines represent reflections along the edges it crosses, and the solid single-arrow lines represent rotations.
• Figure 2: The constellation of the spherical code in Example \ref{['example:224XZ']}. The logical $0$ state is superposed by the states localized at red and pink points, while logical $1$ is by blue and light blue ones. The red and blue points have a coefficient of $1$ while the pink and light blue ones have a coefficient of $-1$.
• Figure 3: The unit cell of the constellation of the Euclidean plane code. The colours represent the same as those in Figure \ref{['fig:224XZsphere']}. The logical operations are implemented by rotation around the corresponding vertices.
• Figure 4: The unit cell of the constellation of the qutrit code in Example \ref{['example:333XZ']}. The red, blue and green points consist of the logical $| {0} \rangle$, $| {1} \rangle$, $| {2} \rangle$ states respectively. The coefficients in the superposition are labelled next to the points. The logical operations are implemented by rotation around the corresponding vertices counterclockwise by $\frac{2\pi}{3}$.
• Figure 5: This figure compares the code constructed from the same encoding map but with a different initial state $| {p_i} \rangle$. The configurations of the spherical code in Example \ref{['example:224XZ']}, Case \ref{['item:224XZflat']} and Case \ref{['item:224XZoptimal']} are shown in subfigures (a), (b) and (c) respectively. The coloring convention is the same as the one used in the main article.