My name is Anastasiia Anishchenko. I am a PhD student at the Theoretical Polymer Physics Department ,

This page is devoted to my research "Continuous-time quantum walks (C
TQW) on different networks" and may be useful for those, who are interested in statistical physics, in quantum mechanics, and, in particular, in the properties of complex networks and of transport over them.

Everything is networked and complexed!

In this world, everything is a network: Internet, relations between people, and even trees. In order to understand the nature, scientists do study networks since many years. A word "network" appears  in scientific papers ranging from biology and chemistry till quantum computations.
During my PhD program study, working
closely with my supervisor Prof. Dr. Oliver Mülken and with the Head of Department Prof. Dr. Alexander Blumen, I try to answer the question: which networks are the most optimal for classical and quantum search algorithms?

During this study, we solve quite a number of problems mainly numerically; besides, quite a number of mathematical problems (linear algebra, chaos theory, probability theory etc). appears:
  • which are the statistical weights of the configurations of a (given) graph?
  • isomorphism detection;
  • how does the spectrum of the Laplacian of  a (given) graph look like? how to generate it numerically?
  • what is the spectral dimension of a certain fractal?


CTQW on complex networks

Transfer processes play a very important role in many physical, chemical or biological instances, involving energy, mass or charge. Such transport processes strongly depend on the topology of the system under study, as, for instance in simple crystals, in complex molecules  or in general networks.  

Classical transport processes modelled by continuous-time random walks (CTRW) are described by a master equation, involving a transfer operator T based on the topology of the system.  A quantum mechanical analogue of CTRW, namely, one variant of the continuous-time quantum walk (CTQW), can be introduced by using the transfer operator  T in defining the Hamiltonian H. For simple lattices this is equivalent to a nearest-neighbor hopping model. The transformation replaces the classical diffusion process by a quantal propagation through the structure. Sufficient conditions for the symmetry of distribution for CTQW in one dimension is are derived in .  CTQW can be related to so-called quantum graphs (QG) .  QG consider bonds, whose length may vary and which may be directed. Therefore, CTQW may be viewed as a simplified version of QG. The connection between discrete Laplacians on descrete QG and periodic orbits were recently studied by Smilansky.