CanaDAM 2017
Université Ryerson, 12 - 16 juin 2017

Algebraic graph theory in quantum computing
Org: Ada Chan (York University)

MARK KEMPTON, Harvard University
Quantum state transfer on graphs  [PDF]

I will discuss perfect and approximately perfect quantum state transfer on graphs. In particular, I will discuss when adding an energy potential to the vertices of a graph can affect whether or not a graph admits perfect or near perfect state transfer. For paths of length at least 4, there is no choice of a potential for which the path admits perfect state transfer, but there is a potential for which paths achieve nearly perfect, or ``pretty good," quantum state transfer. I will discuss this and other related results.

CHRISTOPHER VAN BOMMEL, University of Waterloo
Characterizing Pretty Good State Transfer on Paths  [PDF]

For a continuous-time quantum walk determined by the XY-Hamiltonian on a path graph, previously the only known examples of pretty good state transfer between internal vertices occurred when there was pretty good state transfer between the end vertices. We determine an infinite family of paths for which there is pretty good state transfer between internal vertices but not between the end vertices, and prove that this is the only such family, completely characterizing pretty good state transfer on paths. Includes joint work with Gabriel Coutinho and Krystal Guo.

LUC VINET, University of Montreal

Perfect state transfer (PST) occurs in spin chains with NN couplings. The simplest model is based on the Krawtchouk polynomials. PST can also be realized in photonic lattices where restricting to NN interactions is obviously an approximation.

Fractional revival (FR) happens in certain chains but not in the NN Krawtchouk model. Like PST, FR is useful in quantum information and can generate entanglement.

I shall present an analytic extension of the NN Krawtchouk model with next-to-nearest neighbour couplings. It will be shown to admit PST as well as FR in distinction to the NN situation.

THOMAS WONG, University of Texas at Austin
Degenerate Perturbation Theory as a Tool for Quantum Search  [PDF]

Degenerate perturbation theory is a ``textbook tool'' for quantum mechanics, famously used to derive the spectra of atoms in the presence of an external electric field (i.e., the Stark effect). In this talk, we show that it can also be used to analyze quantum computing algorithms, specifically quantum search on graphs. Using it, we show two intuitions to be false, that global symmetry and high connectivity are not necessary for fast quantum search.

HARMONY ZHAN, University of Waterloo
Discrete-Time Quantum Walks and Graph Structures  [PDF]

A discrete-time quantum walk can be described by successive application of a unitary matrix that incorporates the structure of a graph. Currently there are more than one way to define this unitary operator. We will show how different models are related to different structures of the underlying graph, and talk about some open problems.


Atlantic Association for Research in the Mathematical Sciences Centre de recherches mathmatiques The Fields Institute Pacific Institute for the Mathematical Sciences Socit mathmatique du Canada Université Ryerson Office of Naval Research Science and Technology