  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 graphs (known as diamond fractals)
which support perfect quantum state transfer, and which have a wide range of
different Hausdorff and spectral dimensions. The resulting systems are spin
networks combining Dyson hierarchical model structure with transverse
permutation symmetries of varying order.
