Spectra of Perfect State Transfer Hamiltonians on Fractal-Like Graphs

Abstract

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, spectral properties, and the geometry of the underlying graph, in order to design novel protocols for applications in quantum information science. We present a new lifting and gluing construction, and use this to prove results concerning an inductive spectral structure, applicable to a wide variety of fractal-like graphs. We illustrate this construction with explicit examples for several classes of diamond graphs.

Hamiltonian systems, Toda lattices, Solitons, Lax Pairs on weighted Z-graded graphs

“…In this section, we mostly extend the machinery developed in previous work [DDMT19,MDDT20] to the case of infinite graphs. We follow [Fom94,Sta12,Sta88] and define a Z-graded graph.…”

“…In [DDMT19,MDDT20] we considered graphs with uniform transversal degrees, which is a particular case of the following assumption.…”

“…In [DDMT19,MDDT20] we showed that a certain class of fractal-type graphs demonstrates favorable geometrical properties when used as ambient spaces for quantum systems. Briefly speaking, we considered a transversal layer structure of finite graphs (for more details, see [MDDT20, Definition 2.1]).…”

“…We interpret this 1D chain in a similar manner as a 1D qubit chain, where the spin-spin interaction is described by a given Jacobi matrix. In [DDMT19,MDDT20] we introduced a technique on how to relate the qubit network on the graph and the auxiliary 1D chain. By doing so, we were able to reduce some quantum systems on graphs to one-dimensional solvable models.…”