Gamal Mograby

Tufts University | Tufts · Department of Mathematics

In this paper, we study Grover's search algorithm focusing on continuous-time quantum walk on graphs. We propose an alternative optimization approach to Grover's algorithm on graphs that can be summarized as follows: instead of finding specific graph topologies convenient for the related quantum walk, we fix the graph topology and vary the underlyi...

We introduce and study self-similar versions of the one-dimensional almost Mathieu operators. Our definition is based on a class of self-similar Laplacians [Formula: see text] instead of the standard discrete Laplacian and includes the classical almost Mathieu operators as a particular case, namely, when the Laplacian’s parameter is [Formula: see t...

We present a detailed spectral analysis for a class of fractal-type diamond graphs, referred to as bubble-diamond graphs, and provide a gap-labeling theorem in the sense of Bellissard for the corresponding probabilistic graph Laplacians using the technique of spectral decimation. The spectrum of the natural Laplacian on limit graphs is shown (gener...

We study the spectral theory of a class of piecewise centrosymmetric Jacobi operators defined on an associated family of substitution graphs. Given a finite centrosymmetric matrix viewed as a weight matrix on a finite directed path graph and a probabilistic Laplacian viewed as a weight matrix on a locally finite strongly connected graph, we constru...

We consider the relationship between the Laplacians on two sequences of planar graphs, one from the theory of self-similar groups and one from analysis on fractals. By establishing a spectral decimation map between these sequences we give an elementary calculation of the spectrum of the former, which was first computed by Grigorchuk and \v{S}uni\'{...

We introduce self-similar versions of the one-dimensional almost Mathieu operators. Our definition is based on a class of self-similar Laplacians instead of the standard discrete Laplacian and includes the classical almost Mathieu operators as a particular case. Our main result establishes that the spectra of these self-similar almost Mathieu opera...

We consider discrete one-dimensional nonlinear equations and present the procedure of lifting them to Z-graded graphs. We identify conditions that allow one to lift one-dimensional solutions to solutions on graphs. In particular, we prove the existence of solitons for static potentials on graded fractal graphs. We also show that even for a simple e...

In this paper we study the spectral features, on fractal-like graphs, of Hamiltonians which exhibit the special property of perfect quantum state transfer (PQST): the transmission of quantum states without dissipation. The essential goal is to develop the theoretical framework for understanding the interplay between PQST, spectral properties, and t...

In the quest for designing novel protocols for quantum information and quantum computation, an important goal is to achieve perfect quantum state transfer for systems beyond the well-known one- dimensional cases, such as 1D spin chains. We use methods from fractal analysis and probability to find a new class of quantum spin chains on fractal-like g...

We consider discrete one dimensional nonlinear equations and present the procedure of lifting them to Z-graded graphs. We identify conditions which allow one to lift one dimensional solutions to solutions on graphs. In particular, we prove the existence of solitons {for static potentials} on graded fractal graphs. We also show that even for a simpl...

The primary goal of my thesis is to study the interplay between properties of physical systems (mostly for quantum information processing) and the geometry of these systems. The ambient spaces I have been working with are fractal-type graphs. In many cases analytic computations can be done on these graphs due to their self-similarity. Different sce...

We consider criteria for the differentiability of functions with continuous Laplacian on the Sierpiński Gasket and its higher-dimensional variants [Formula: see text], [Formula: see text], proving results that generalize those of Teplyaev [Gradients on fractals, J. Funct. Anal. 174(1) (2000) 128–154]. When [Formula: see text] is equipped with the s...

In this paper we study the spectral features, on fractal-like graphs, of Hamiltonians which exhibit the special property of perfect quantum state transfer: the transmission of quantum states without dissipation. The essential goal is to develop the theoretical framework for understanding the interplay between perfect quantum state transfer, spectra...

We study the discretization of a Dirichlet form on the Koch snowflake domain and its boundary with the property that both the interior and the boundary can support positive energy. We compute eigenvalues and eigenfunctions, and demonstrate the localization of high energy eigenfunctions on the boundary via a modification of an argument of Filoche an...

We extend the analysis of perfect quantum state transfer beyond one dimensional spin chains to show that it can be achieved and designed on a large class of fractal structures, known as diamond fractals, which have a wide range of Hausdorff and spectral dimensions. The resulting systems are spin networks combining Dyson hierarchical model structure...

We consider criteria for the differentiability of functions with continuous Laplacian on the Sierpinski Gasket and its higher-dimensional variants $SG_N$, $N>3$, proving results that generalize those of Teplyaev. When $SG_N$ is equipped with the standard Dirichlet form and measure $\mu$ we show there is a full $\mu$-measure set on which continuity...