Quantum Information and Computing
A.Y. 2026/2027
Learning objectives
The course introduces the basics of the quantum paradigm and its application to quantum computing. The student will discover main differences between the classical and quantum paradigm of computation, consolidating knowledge both in the suitable mathematical framework and by designing quantum software.
Expected learning outcomes
The student will be able to understand the foundations of quantum computing and the motivations for its introduction in the classical computing realm. She/He will acquire the ability to understand and use main techniques for the design and analysis of quantum algorithms, presented by investigating some paradigmatic examples. She/He will also be able to deep more advanced topics, as well as provide a software implementation of quantum algorithms.
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
Introduction
· Historical perspective of quantum computing, main events
- quantum mechanics
- computer science
- information theory
Quantum computing, a first invitation: qbit
· From classical to quantum computing
- classical bits and circuits
- qbit as superposition of basis states, mathematical definition
- qbit observation, probabilistic information extraction
· Some ideas on physical realizations of qbits, double-slit experiment
Mathematical foundations of quantum mechanics and 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
- projectors and outer product
· 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
Quantum mechanics postulates and quantum computational systems
· A formal description of:
- qbit, qregister, observable
- observables on qregisters: canonical observable and other observables
- non-locality in quantum mechanics
- entangled (or EPR, or Bell) states
- hidden information in quantum systems, hints of quantum superiority
· Quantum circuits, analogies with the classical case
- single qbit quantum gates: X, Z, H, rotations, Pauli matrices
- linear-algebraic and geometric properies of single qbit quantum gates
- complete basis for single qbit quantum gates
- quantum coin tossing by H: elements of quantum walks
- quantum vs. probabilistic dynamics: constructive/destructive interference
- quantum circuit analysis: on basis/matrix/wire-per-wire/bit-per-bit
- two-qbit quantum gates: controlled-NOT, main properties
- complete basis for quantum gates
- implementation of a circuit for quantum SWAP
- concrete circuit production of entangled states
- examples of design and analysis of quantum circuits
- measurement gates, functioning, resulting probabilistic dynamics
- conditional quantum operation implementations, "quantum if"
- quantum >= classical paradigm: a circuit perspective
- quantum parallelism
Introduction to quantum algorithms
· Bloch sphere for qbit representation
· No-cloning theorem
· Using entanglement for:
- quantum teleportation
- superdense coding
· Reversible computations
- Toffoli gate
- NAND via Toffoli gate
· 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
Final seminars on advanced topics
· Historical perspective of quantum computing, main events
- quantum mechanics
- computer science
- information theory
Quantum computing, a first invitation: qbit
· From classical to quantum computing
- classical bits and circuits
- qbit as superposition of basis states, mathematical definition
- qbit observation, probabilistic information extraction
· Some ideas on physical realizations of qbits, double-slit experiment
Mathematical foundations of quantum mechanics and 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
- projectors and outer product
· 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
Quantum mechanics postulates and quantum computational systems
· A formal description of:
- qbit, qregister, observable
- observables on qregisters: canonical observable and other observables
- non-locality in quantum mechanics
- entangled (or EPR, or Bell) states
- hidden information in quantum systems, hints of quantum superiority
· Quantum circuits, analogies with the classical case
- single qbit quantum gates: X, Z, H, rotations, Pauli matrices
- linear-algebraic and geometric properies of single qbit quantum gates
- complete basis for single qbit quantum gates
- quantum coin tossing by H: elements of quantum walks
- quantum vs. probabilistic dynamics: constructive/destructive interference
- quantum circuit analysis: on basis/matrix/wire-per-wire/bit-per-bit
- two-qbit quantum gates: controlled-NOT, main properties
- complete basis for quantum gates
- implementation of a circuit for quantum SWAP
- concrete circuit production of entangled states
- examples of design and analysis of quantum circuits
- measurement gates, functioning, resulting probabilistic dynamics
- conditional quantum operation implementations, "quantum if"
- quantum >= classical paradigm: a circuit perspective
- quantum parallelism
Introduction to quantum algorithms
· Bloch sphere for qbit representation
· No-cloning theorem
· Using entanglement for:
- quantum teleportation
- superdense coding
· Reversible computations
- Toffoli gate
- NAND via Toffoli gate
· 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
Final seminars on advanced topics
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
Professor(s)
Reception:
On appointment, via email
Room S 6008, VI floor, Dip. Informatica "Giovanni Degli Antoni", via Celoria 18, 20133 Milano, Italy