Self-Similar Algebras with connections to Run-length Encoding and Rational Languages

A self-similar algebra $\left(\mathfrak{A}, \psi \right)$ is an associative algebra $\mathfrak{A}$ with a morphism of algebras $\psi: \mathfrak{A} \longrightarrow M_d \left( \mathfrak{A}\right)$, where $M_d \left( \mathfrak{A}\right)$ is the set of $d\times d$ matrices with coefficients from $\mathfrak{A}$. We study the connection between self-similar algebras with run-length encoding and rational languages. In particular, we provide a curious relationship between the eigenvalues of a sequence of matrices related to a specific self-similar algebra and the smooth words over a 2-letter alphabet. We also consider the language $L(s)$ of words $u$ in $(\Sigma\times \Sigma)^*$ where $\Sigma=\{0,1\}$ such that $s\cdot u$ is a unit in $\mathfrak{A}$. We prove that $L(s)$ is rational and provide an asymptotic formula for the number of words of a given length in $L(s)$

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

Experimental quantum key distribution with finite-key security analysis for noisy channels

In quantum key distribution implementations, each session is typically chosen long enough so that the secret key rate approaches its asymptotic limit. However, this choice may be constrained by the physical scenario, as in the perspective use with satellites, where the passage of one terminal over the other is restricted to a few minutes. Here we demonstrate experimentally the extraction of secure keys leveraging an optimal design of the prepare-and-measure scheme, according to recent finite-key theoretical tight-bounds. The experiment is performed in different channel conditions, and assuming two distinct attack models: individual attacks, or general quantum attacks. The request on the number of exchanged qubits is then obtained as a function of the key size and of the ambient quantum bit error rate. The results indicate that viable conditions for effective symmetric, and even one-time-pad, cryptography are achievable.Comment: 20 pages, 4 figure