Quantum Walks

A.Y. 2021/2022
Overall hours
Learning objectives
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.
Expected learning outcomes
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.
Course syllabus and organization

Single session

Lesson period
Second semester
More specific information on the delivery modes of training activities for academic year 2021/22 will be provided over the coming months, based on the evolution of the public health situation.
Course syllabus
I. Introduction
o Review of probability theory and stochastic processes
o Discrete- and continuous-time classical random walks
o Discrete- and continuous-time quantum walks
o Introduction to graph theory
o Quantum walks on graphs: ring, complete and star graphs

II. Applications
o Graph crossing and decision trees
o Quantum spatial search by quantum walks
o Dimensional reduction
o Perfect state transfer protocol
o Quantum PageRank

III. Beyond the single particle model
o Formalism: Hilbert and Fock space
o Two-particle Hubbard model

IV. Experimental implementations and decoherence
o Review of some recent experimental platforms for quantum walks
o Disorder: Anderson Localization
o Noise: Stochastic fluctuations and decoherence
o Numerical simulation (Python, Mathematica,..) of quantum walks.
Prerequisites for admission
The course is structured to be self-consistent. Students should know the basics of quantum mechanics and linear algebra.
Teaching methods
Theoretical lectures at the blackboard, supported by slides for the dynamical contents.
Teaching Resources
-Renato Portugal, Quantum Walks and Search Algorithms, Springer (2018)
- Kia Manouchehri and Jingbo Wang, Physical Implementation of Quantum Walks, Springer (2014)
- lecture notes and slides
Assessment methods and Criteria
The exam is an oral interview (lasting about from 45 to 75 minutes) in which both the knowledge acquired during the lectures and the critical skills about analyzing problems related to the same topics will be assessed.
FIS/03 - PHYSICS OF MATTER - University credits: 6
Lessons: 42 hours
Professor: Benedetti Claudia
Educational website(s)