Fractal spectral triples on Kellendonk's C∗C^*-algebra of a substitution tiling

We introduce a new class of noncommutative spectral triples on Kellendonk's $C^*$-algebra associated with a nonperiodic substitution tiling. These spectral triples are constructed from fractal trees on tilings, which define a geodesic distance between any two tiles in the tiling. Since fractals typically have infinite Euclidean length, the geodesic distance is defined using Perron-Frobenius theory, and is self-similar with scaling factor given by the Perron-Frobenius eigenvalue. We show that each spectral triple is $\theta$-summable, and respects the hierarchy of the substitution system. To elucidate our results, we construct a fractal tree on the Penrose tiling, and explicitly show how it gives rise to a collection of spectral triples.Comment: Updated to agree with published versio

Constructions and Noise Threshold of Hyperbolic Surface Codes

We show how to obtain concrete constructions of homological quantum codes based on tilings of 2D surfaces with constant negative curvature (hyperbolic surfaces). This construction results in two-dimensional quantum codes whose tradeoff of encoding rate versus protection is more favorable than for the surface code. These surface codes would require variable length connections between qubits, as determined by the hyperbolic geometry. We provide numerical estimates of the value of the noise threshold and logical error probability of these codes against independent X or Z noise, assuming noise-free error correction

Taylor-Socolar hexagonal tilings as model sets

The Taylor-Socolar tilings are regular hexagonal tilings of the plane but are distinguished in being comprised of hexagons of two colors in an aperiodic way. We place the Taylor-Socolar tilings into an algebraic setting which allows one to see them directly as model sets and to understand the corresponding tiling hull along with its generic and singular parts. Although the tilings were originally obtained by matching rules and by substitution, our approach sets the tilings into the framework of a cut and project scheme and studies how the tilings relate to the corresponding internal space. The centers of the entire set of tiles of one tiling form a lattice $Q$ in the plane. If $X_Q$ denotes the set of all Taylor-Socolar tilings with centers on $Q$ then $X_Q$ forms a natural hull under the standard local topology of hulls and is a dynamical system for the action of $Q$. The $Q$-adic completion $\bar{Q}$ of $Q$ is a natural factor of $X_Q$ and the natural mapping $X_Q \longrightarrow \bar{Q}$ is bijective except at a dense set of points of measure 0 in $\bar{Q}$. We show that $X_Q$ consists of three LI classes under translation. Two of these LI classes are very small, namely countable $Q$-orbits in $X_Q$. The other is a minimal dynamical system which maps surjectively to $\bar{Q}$ and which is variously $2:1$, $6:1$, and $12:1$ at the singular points. We further develop the formula of Socolar and Taylor (2011) that determines the parity of the tiles of a tiling in terms of the co-ordinates of its tile centers. Finally we show that the hull of the parity tilings can be identified with the hull $X_Q$; more precisely the two hulls are mutually locally derivable.Comment: 45 pages, 33 figure