Quantum Walks

The aim of the course is to provide students with the knowledge and tools for the theoretical study of quantum walks (QW). The concepts of continuous- and discrete- time QW will be studied on graphs of different topology. Some of the most important applications of QW will be discussed, such as the quantum spatial search algorithm and the protocol for the perfect transfer of quantum. The generalization of many-particle QW will be introduced and recent experimental implementations of QW will be presented.

At the end of the course the student will be able to:
1.Use the mathematical formalism to describe a continuous- and discrete-time quantum walks and discuss the main differences with their classical analogues.
2.Characterize quantum walks on graphs of different topology
3.Describe the main applications of QW in the context of algorithms, communication and transport.
4.Use dimensional reduction techniques where possible and convenient
5.List the necessary and sufficient conditions so that it is possible to perform a perfect transfer of quantum states using the QW formalism
6.Generalize the concept of QW to many particles. In particular, they will be able to analytically solve the problem of two particles described by Hubbard Hamiltonian
7.Present the main experimental platforms for QW and discuss problems related to the sources of noise and decoherence.