97263 - Mathematical Foundations of Quantum Computation

Course Unit Page

Academic Year 2021/2022

Learning outcomes

At the end of the course the student is acquainted with the basic aspects of quantum computation like those of entanglement, qubits, quantum algorithms and information. He is able to deal with the classical problems of the subject.

Course contents

Prerequisites

The basic courses of the BSc (Laurea Triennale) in Mathematics. Knowledge of linear algebra is particularly important.

Syllabus

Introduction and overview

Mathematical foundations of quantum mechanics:
Quantum states and density operators
Qubits and the Bloch sphere
Composite quantum systems
Quantum measurements and quantum operations

Entanglement theory:
Entanglement
The EPR paradox
Bell's theorem
The no-cloning theorem
Quantum teleportation
Superdense coding

Classical computation:
Turing machines
Complexity classes
The gate model
Universal gate sets
Reversible computation

Quantum computation:
One-qubit and two-qubit gates
Universal gate sets
The Gottesman-Knill theorem

Quantum algorithms:
The Deutch-Jozsa algorithm
The Berenstein-Vazirani algorithm
Simon's algorithm
The quantum Fourier transfrom
The period-finding algorithm
Shor's factoring algorithm
Grover's search algorithm

Quantum error correction:
The 3-qubit code
The 9-qubit code
The Knill-Laflamme theorem
CSS codes

Classical cryptography:
Private key vs. public key
One-time pad
RSA
Breaking RSA with Shor's algorithm

Quantum cryptography:
Quantum key distribution
BB84 protocol

Readings/Bibliography

The lectures will mostly follow the book

Michael A. Nielsen, Isaac L. Chuang, "Quantum Computation and Quantum Information", Cambridge University Press, 2010

A free further resource available online is

John Preskill, "Quantum Computation" (lecture notes), http://theory.caltech.edu/~preskill/ph229/

Teaching methods

Classroom lectures

Assessment methods

The grade will be assigned upon oral examination. Each exam will include theoretical questions and the solution of a simple exercise. The student will be evaluated both on the understanding of the content of the lectures and on the ability to apply it. The students who achieve an organic vision of the topics, are able to critically apply them and master the specific language will be assessed with a grade of excellence. The students who show a mechanic or mnemonic knowledge of the topics, have limited capabilities of analysis and synthesis or a correct but not always appropriate language will be assessed with an average grade. The students who show a minimal knowledge of the topics and of the appropriate language will be assessed with the minimum passing grade. The students who show poor knowledge of the topics, inappropriate language and are not able to properly understand the material will be assessed with a non-passing grade.

Office hours

See the website of Giacomo De Palma