Combinatorial and algebraic aspects of quantum state transfer
Xiaohong Zhang
Reliably transferring a quantum state from one location to another, as well as generating entangled states, are important tasks to achieve in quantum spin systems. The fidelity or probability of state transfer is a number between 0 and 1 that measures the closeness of two quantum states. Fidelity is used to determine the accuracy of quantum state transfer. There are several interesting phenomena of quantum state transfer defined via fidelity: perfect state transfer, pretty good state transfer, and fractional revival. This thesis contains results about the perfect state transfer property of some special classes of graphs, including Hadamard diagonalizable graphs, weighted paths with loops, as well as switched and partially switched hypercubes. A correspondence between the class of graphs that are diagonalizable by a standard Hadamard matrix and the class of cubelike graphs is given. Sensitivity of fidelity to errors when perfect state transfer occurs is analysed: if a system admits perfect state transfer at some time t, bounds on fidelity of state transfer at t+h for very small h are given, as well as bounds on fidelity of a slightly perturbed system at time t. Finally, Laplacian fractional revival on graphs is considered; in particular the thesis contains a characterization of threshold graphs that admit Laplacian fractional revival.
•Aug 29, 2019•0 citations