Perfect quantum state transfer on diamond fractal graphs

Abstract

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 with transverse permutation symmetries of varying order.

Spectra of Perfect State Transfer Hamiltonians on Fractal-Like Graphs

“…In this section, we extend the study of PQST on diamond fractal graphs [DDMT19] to a more general class of graphs. Let G = (V (G), E(G)) be a finite connected graph with a vertex set V (G) and an edge set E(G).…”

“…The following definition generalizes the concept of the intrinsically transversal layers introduced in [DDMT19]. This concept can be found in [HO07, page 76] under the name stratification and plays a crucial role in the quantum decomposition of a graph adjacency matrix.…”

“…A Hamiltonian on G is a self-adjoint operator H acting on L 2 (G). It was observed in [DDMT19] that constructing a Hamiltonian, which is not only adapted to the graph structure but also to the given transversal decomposition of the diamond-type graphs, leads indeed to a Hamiltonian, that realizes a perfect quantum state transfer. Motivated by these observations, we impose the following assumptions on H: Assumption 2.7 (Assumptions on the Hamiltonian).…”

“…These works illustrate the fact that PQST is a rare phenomenon, for which the construction of explicit examples remains rather non-trivial. Intending to investigate the rich interplay between quantum state transfer and geometries beyond one-dimensional graphs, we showed in a previous paper [DDMT19] that PQST is possible on the large and diverse class of fractal-type diamond graphs. A significant interest in these graphs lies in the fact that their limit spaces constitute a family of fractals, which present different geometrical properties, including a wide range of Hausdorff and spectral dimensions.…”

“…In this paper, we generalize the construction in [DDMT19] and show that it works for any graph possessing a transversal decomposition (see assumption 2.10). More precisely, on such a graph, a Hamiltonian based on nearest-neighbor coupling and with a certain transversal projective structure (see assumptions 2.7) can be engineered to admit a PQST.…”