Quantum Walks
A.Y. 2024/2025
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.
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.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
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 graph
II. Applications
o Graph crossing and decision trees
o Quantum spatial search by quantum walks
o Perturbation theory and dimensional reduction
o Perfect state transfer protocol
o Quantum cryptography assisted by discrete-time quantum walks
III. Beyond the single particle model
o Two-particle Hubbard model
o Scattering and bound states
IV. Experimental implementations and decoherence
o Review of some recent experimental platforms for quantum walks
o Noise: Stochastic fluctuations and decoherence
o Numerical simulation (Python, Mathematica,..) of quantum walks.
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 graph
II. Applications
o Graph crossing and decision trees
o Quantum spatial search by quantum walks
o Perturbation theory and dimensional reduction
o Perfect state transfer protocol
o Quantum cryptography assisted by discrete-time quantum walks
III. Beyond the single particle model
o Two-particle Hubbard model
o Scattering and bound states
IV. Experimental implementations and decoherence
o Review of some recent experimental platforms for quantum walks
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)
- Scientific publications
- lecture notes and slides
- Scientific publications
- 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.
Professor(s)