Quantum Information and Computing
A.Y. 2025/2026
Learning objectives
The course introduces the basics of the quantum paradigm and its application to the field of computing. The student will discover main differences between the classical and quantum paradigm, consolidating knowledge both in the suitable mathematical framework and by designing quantum software.
Expected learning outcomes
The student will be able to understand the premises of the quantum paradigm and the motivations for its application within different fields of computer science. She/He will acquire the ability to solve elementary problems in quantum mechanics that are of interest for the physical implementation of quantum devices for communication and computation. She/He will also learn the working principles of some paradigmatic algorithms of quantum key distribution and of some of the most important algorithms in quantum computing.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
Introduction
· Historical perspective of quantum computing, main events
- quantum mechanics
- computer science
- information theory
Qbit
· From classical to quantum computing
- classical bits and circuits
- qbit as superposition of basis states, mathematical definition
- qbit measuring, probabilistic information extraction
- Bloch sphere representation for qbits
· Hints on physical realizations of qbits, double-slit experiment
Qregister and quantum circuits
· Qregister
- mathematical definition, observation
- entangled states (Bell, EPR)
· Quantum circuits, part I
- quantum gates, mathematical and geometric interpretation
- single qbit quantum gates: Pauli matrices, H (Hadamard)
- multiple qbits quantum gates: controlled-NOT
- entangled state generation
- SWAP quantum circuit
- measurement gates
- design and analysis of quantum circuits
Mathematical foundations of quantum computing
· Complex numbers
- cartesian, polar, and trigonometric form
- conjugate, modulo, operations
- exponential representation and Euler's identity
· Vector spaces
- orthonormal basis and vector subspaces
- inner product and Hilbert spaces
- Dirac notation, bra, ket, outer product
· Linear operators and matrices
- eigenvalues, eigenvectors, eigenspaces
- normal, Hermitian, unitary matrices
- spectral decomposition for normal matrices
· Tensor product of spaces and matrices (Kronecker)
Quantum mechanics
· Postulates
- postulate 1: Hilbert space as state space
- postulate 2: unitary operator as dynamics, Schrödinger equation
- postulate 3: Hermitian matrix as observable, wave function collapse
- postulate 4: tensor product as quantum system composition
· Application of postulates in the description of quantum circuits
Quantum circuits, part II
· No-cloning theorem
· Using entanglement
- quantum teleportation
- superdense coding
Quantum algorithms introduction
· Reversible computations
- Toffoli gate
- NAND via Toffoli
· Quantum parallelism
- Deutsch algorithm
- Deutsch-Josza algorithm
Quantum search algorithm
· Grover's algorithm
- oracle and diffusion operator
- geometric interpretation
- time complexity analysis
- a practical example
Quantum Fourier transform (QFT) and its applications
· Phase estimation for unitary matrix eigenvalues
· Quantum order finding
- modular matrix
- ket 1 as eigenvector
- continued fractions and order from convergents
· Factoring: Shor's algorithm
- Euclidean algorithm and factors
- non-trivial factors from quadratic congruences
- confidence estimation
- time complexity analysis
· Historical perspective of quantum computing, main events
- quantum mechanics
- computer science
- information theory
Qbit
· From classical to quantum computing
- classical bits and circuits
- qbit as superposition of basis states, mathematical definition
- qbit measuring, probabilistic information extraction
- Bloch sphere representation for qbits
· Hints on physical realizations of qbits, double-slit experiment
Qregister and quantum circuits
· Qregister
- mathematical definition, observation
- entangled states (Bell, EPR)
· Quantum circuits, part I
- quantum gates, mathematical and geometric interpretation
- single qbit quantum gates: Pauli matrices, H (Hadamard)
- multiple qbits quantum gates: controlled-NOT
- entangled state generation
- SWAP quantum circuit
- measurement gates
- design and analysis of quantum circuits
Mathematical foundations of quantum computing
· Complex numbers
- cartesian, polar, and trigonometric form
- conjugate, modulo, operations
- exponential representation and Euler's identity
· Vector spaces
- orthonormal basis and vector subspaces
- inner product and Hilbert spaces
- Dirac notation, bra, ket, outer product
· Linear operators and matrices
- eigenvalues, eigenvectors, eigenspaces
- normal, Hermitian, unitary matrices
- spectral decomposition for normal matrices
· Tensor product of spaces and matrices (Kronecker)
Quantum mechanics
· Postulates
- postulate 1: Hilbert space as state space
- postulate 2: unitary operator as dynamics, Schrödinger equation
- postulate 3: Hermitian matrix as observable, wave function collapse
- postulate 4: tensor product as quantum system composition
· Application of postulates in the description of quantum circuits
Quantum circuits, part II
· No-cloning theorem
· Using entanglement
- quantum teleportation
- superdense coding
Quantum algorithms introduction
· Reversible computations
- Toffoli gate
- NAND via Toffoli
· Quantum parallelism
- Deutsch algorithm
- Deutsch-Josza algorithm
Quantum search algorithm
· Grover's algorithm
- oracle and diffusion operator
- geometric interpretation
- time complexity analysis
- a practical example
Quantum Fourier transform (QFT) and its applications
· Phase estimation for unitary matrix eigenvalues
· Quantum order finding
- modular matrix
- ket 1 as eigenvector
- continued fractions and order from convergents
· Factoring: Shor's algorithm
- Euclidean algorithm and factors
- non-trivial factors from quadratic congruences
- confidence estimation
- time complexity analysis
Prerequisites for admission
Mathematical and algorithmic skills typically acquired after the first two years. Familiarity with elementary topics of linear algebra and probability is welcome.
Teaching methods
The course basically consists of traditional lectures, aiming to introduce to quantum computing from both a theoretical and a practical viewpoint. First, the theoretical foundations of quantum computing will be introduced, within the suitable mathematical background. Then, some advanced seminars will be proposed, dealing with more practical issues. This combined theoretical/practical approach will enable to consolidate knowledge on the quantum paradigm.
Teaching Resources
Handouts and other material: available at course website
· Slides and handouts by instructors.
Textbooks: both available online at UniMi library:
· M.A. Nielsen, I.L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2010.
· S. Olivares. A Student's Guide to Quantum Computing. Springer, 2025.
Websites
· course website on myAriel university platform
· Qibo, a middleware for quantum software design: https://qibo.science
· Slides and handouts by instructors.
Textbooks: both available online at UniMi library:
· M.A. Nielsen, I.L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2010.
· S. Olivares. A Student's Guide to Quantum Computing. Springer, 2025.
Websites
· course website on myAriel university platform
· Qibo, a middleware for quantum software design: https://qibo.science
Assessment methods and Criteria
The student is allowed to choose between either a written or an oral examination; in case of written examination, two hours are devoted to the test. Regardless the form, the examination aims at testing the knowledge of theoretical and algorithmic bases of quantum computing. The exam will be rated from 1 to 30. An evaluation between 18 and 23 indicates an appropriate level of knowledge of basics of quantum computing, an evaluation between 24 and 27 indicates a good knowledge of such basics, higher evaluations indicate a very good knowledge and originality in applying basics of quantum computing.
INF/01 - INFORMATICS - University credits: 6
Lessons: 48 hours
Professors:
Mereghetti Carlo, Palano Beatrice Santa
Educational website(s)
Professor(s)
Reception:
On appointment, via email
Room S 6008, VI floor, Dip. Informatica "Giovanni Degli Antoni", via Celoria 18, 20133 Milano, Italy