Minimal chordal sense of direction and circulant graphs

A sense of direction is an edge labeling on graphs that follows a globally consistent scheme and is known to considerably reduce the complexity of several distributed problems. In this paper, we study a particular instance of sense of direction, called a chordal sense of direction (CSD). In special, we identify the class of k-regular graphs that admit a CSD with exactly k labels (a minimal CSD). We prove that connected graphs in this class are Hamiltonian and that the class is equivalent to that of circulant graphs, presenting an efficient (polynomial-time) way of recognizing it when the graphs' degree k is fixed

On Self-Dual Quantum Codes, Graphs, and Boolean Functions

A short introduction to quantum error correction is given, and it is shown that zero-dimensional quantum codes can be represented as self-dual additive codes over GF(4) and also as graphs. We show that graphs representing several such codes with high minimum distance can be described as nested regular graphs having minimum regular vertex degree and containing long cycles. Two graphs correspond to equivalent quantum codes if they are related by a sequence of local complementations. We use this operation to generate orbits of graphs, and thus classify all inequivalent self-dual additive codes over GF(4) of length up to 12, where previously only all codes of length up to 9 were known. We show that these codes can be interpreted as quadratic Boolean functions, and we define non-quadratic quantum codes, corresponding to Boolean functions of higher degree. We look at various cryptographic properties of Boolean functions, in particular the propagation criteria. The new aperiodic propagation criterion (APC) and the APC distance are then defined. We show that the distance of a zero-dimensional quantum code is equal to the APC distance of the corresponding Boolean function. Orbits of Boolean functions with respect to the {I,H,N}^n transform set are generated. We also study the peak-to-average power ratio with respect to the {I,H,N}^n transform set (PAR_IHN), and prove that PAR_IHN of a quadratic Boolean function is related to the size of the maximum independent set over the corresponding orbit of graphs. A construction technique for non-quadratic Boolean functions with low PAR_IHN is proposed. It is finally shown that both PAR_IHN and APC distance can be interpreted as partial entanglement measures.Comment: Master's thesis. 105 pages, 33 figure

Toeplitz matrices for the long-range Kitaev model

In questa tesi discuteremo delle fasi topologiche di una catena quantistica unidimensionale con accoppiamento superconduttivo, nota anche come catena di Kitaev, insieme a un paio di estensioni di essa: una con accoppiamento a lungo raggio e una con accoppiamento ai bordi della catena. Queste fasi verranno investigate con l'aiuto della teoria delle matrici di Toeplitz, che semplifica sia la risoluzione dello spettro che delle funzioni di correlazione. Inoltre, all'interno della teoria delle matrici di Toeplitz identificheremo un winding number particolare, che potrà essere usato come strumento per rilevare fasi topologiche e edge state non massivi. Sulla base di questa identificazione, insieme ad alcune analisi numeriche eseguite sulla catena di Kitaev a lungo-raggio, proporremo una congettura sulla comparsa di edge state massivi, che verrà usata poi per spiegare una transizione di fase senza chiusura del gap che avviene nella catena di Kitaev a lungo raggio

Linear Time Split Decomposition Revisited

International audienceGiven a family $\mathcal{F}$ of subsets of a ground set V, its orthogonal is defined to be the family of subsets that do not overlap any element of $\mathcal{F}$. Using this tool we revisit the problem of designing a simple linear time algorithm for undirected graph split (also known as 1-join) decomposition