01 / Teaching

Quantum and Post-Quantum Computing.

University of Milan · Academic year 2026/2027

The course descriptions are in English. The downloadable slides are the original files and retain their source language.

Learning path

The seven modules lead from physical foundations and mathematical tools to quantum circuits, algorithms and protocols, with slides available to read and download as PDFs and links in each module to related activities in Quantum Lab.

Starting from the simulation of physical systems and the mathematical tools used to describe them, the materials introduce quantum gates and circuits, on which the Deutsch–Jozsa, Bernstein–Vazirani, Simon, Shor and Grover algorithms are built; the course continues with BB84, superdense coding and error correction, before examining the operational limits of quantum key distribution and its relationship with post-quantum cryptography.

7modules

1,950PDF pages

PDFopen access

You can follow the modules in order, moving from phenomena to mathematical tools and then to circuits, algorithms and protocols, or use the index to return to a topic you wish to revisit.

Quantum and post-quantum

The Bloch sphere describes the pure states of a qubit, while lattices provide the mathematical structure underlying some post-quantum cryptographic constructions, linking two fields that address different problems.

Downloadable modules

Course materials

Module 01

Physical foundations and simulation of quantum systems

From classical and statistical simulation to waves, interference, two-alternative states and composite systems. The module addresses the loss of interference, information and irreversibility, leading to the transition from quantum mechanics to computation.

Classical and statistical simulation
Waves, interference and measurement
Qubits, composite systems and the environment
Information and the construction of a quantum computer

519 PDF pages · 3.1 MB · Marzio De Corato; Danilo Bruschi

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Reading task

Following the progression from simulations to waves and two-alternative states, identify where relative phase changes observed probabilities and how interaction with the environment suppresses interference.

Suggested preparation

Function graphs, ratios and elementary probability are useful starting points; the next module revisits the algebraic tools.

Topic map · 12

Material in Italian · original section titles

Destinations follow bookmarks in the original documents; the number refers to the physical PDF page, which may differ from the printed slide number.

Study → experiment

After reading the slides, you can open these activities to compare the module’s topics with the laboratory’s representations.

Opens in a new tab. Laboratory interface: Italian.

Licence stated in the document: CC BY-NC-SA 4.0.

Module 02

Mathematical foundations: linear algebra, probability and measurement

Complex numbers, vectors, matrices and tensor products establish the language of computation; probability, states and measurement connect mathematical transformations to observable outcomes.

Vectors and linear transformations
Tensor products and matrix functions
Probability and data
Quantum states and measurement

514 PDF pages · 2.7 MB · Marzio De Corato; Danilo Bruschi

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Reading task

After working with complex numbers, vectors and operators, reconstruct a projective measurement, specifying its basis, outcome probabilities and state update.

Suggested preparation

The first module can be read alongside the linear algebra and probability review in the opening sections of these slides.

Topic map · 31

Material in Italian · original section titles

Destinations follow bookmarks in the original documents; the number refers to the physical PDF page, which may differ from the printed slide number.

Study → experiment

After reading the slides, you can open these activities to compare the module’s topics with the laboratory’s representations.

Opens in a new tab. Laboratory interface: Italian.

Licence stated in the document: CC BY-NC-SA 4.0.

Module 03

Quantum circuits: gates, reversibility and the Deutsch and Deutsch–Jozsa algorithms

Reading and executing circuits, single- and multi-qubit gates, composition, phases and interference. Reversible logic leads to the Deutsch and Deutsch–Jozsa algorithms, developed through examples and checks.

Single-qubit gates and composition
Phases, interference and registers
CNOT, Toffoli and ancillary qubits
Deutsch and Deutsch–Jozsa

177 PDF pages · 1.0 MB · Marzio De Corato; Danilo Bruschi

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Reading task

Read a circuit from left to right, following the register through its gates and comparing the role of phases in Deutsch’s derivation and its Deutsch–Jozsa generalisation.

Suggested preparation

Revisit tensor products, unitary operators, the Born rule and changes of basis in module two.

Topic map · 13

Material in Italian · original section titles

Destinations follow bookmarks in the original documents; the number refers to the physical PDF page, which may differ from the printed slide number.

Study → experiment

After reading the slides, you can open these activities to compare the module’s topics with the laboratory’s representations.

Opens in a new tab. Laboratory interface: Italian.

Licence stated in the document: CC BY-NC-SA 4.0.

Module 04

Quantum algorithms: Bernstein–Vazirani and Simon

Two hidden-string reconstruction problems: the coefficients of a binary parity and the XOR relation between inputs with the same output. Oracles, interference and measurement are followed through to reconstruction of the answer.

Binary parities and hidden strings
Oracles and phase marking
Bernstein–Vazirani
Simon and binary constraints

179 PDF pages · 0.9 MB · Marzio De Corato; Danilo Bruschi

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Reading task

Compare Bernstein–Vazirani and Simon through their oracles and measurement information, distinguishing query counts from the classical work needed to interpret outcomes.

Suggested preparation

Circuit reading from module three, bit arithmetic, binary inner products and linear equations are useful preparation.

Topic map · 6

Material in Italian · original section titles

Destinations follow bookmarks in the original documents; the number refers to the physical PDF page, which may differ from the printed slide number.

Study → experiment

After reading the slides, you can open these activities to compare the module’s topics with the laboratory’s representations.

Opens in a new tab. Laboratory interface: Italian.

Licence stated in the document: CC BY-NC-SA 4.0.

Module 05

Shor’s algorithm: modular arithmetic, RSA and the Fourier transform

Divisors, congruences, inverses and modular powers prepare the construction of RSA. Quantum period finding and the Fourier transform lead to message recovery; a final discussion connects the period to factorisation.

Modular arithmetic and Euclid’s algorithm
RSA and the period of the ciphertext
Fourier transform and period finding
From measurement to decryption and factorisation

208 PDF pages · 1.5 MB

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Reading task

Follow the chain linking modular arithmetic, period finding and information recovery, distinguishing the Fourier transform from classical measurement processing.

Suggested preparation

Revisit quantum circuits and complex numbers; divisibility, congruences and order are developed in the opening sections.

Topic map · 13

Material in Italian · original section titles

Destinations follow bookmarks in the original documents; the number refers to the physical PDF page, which may differ from the printed slide number.

Study → experiment

After reading the slides, you can open these activities to compare the module’s topics with the laboratory’s representations.

Opens in a new tab. Laboratory interface: Italian.

Module 06

Grover’s algorithm and amplitude amplification

The search for valid candidates is developed through oracles, reflections and rotations in state space. The examples introduce multiple solutions, an unknown number of solutions and amplitude amplification, keeping costs and limitations explicit.

Search, verifiers and queries
Oracle, diffuser and Grover rotation
Multiple solutions and unknown solution counts
Amplitude amplification and resource analysis

253 PDF pages · 1.1 MB

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Reading task

Interpret Grover iterations as rotations in the relevant subspace and examine how the number of solutions determines when amplification should stop.

Suggested preparation

Revisit state normalisation, reversible oracles and reflections; four- and eight-candidate examples introduce the geometric construction.

Topic map · 12

Material in Italian · original section titles

Destinations follow bookmarks in the original documents; the number refers to the physical PDF page, which may differ from the printed slide number.

Study → experiment

After reading the slides, you can open these activities to compare the module’s topics with the laboratory’s representations.

Opens in a new tab. Laboratory interface: Italian.

Module 07

Quantum protocols: BB84, QKD, superdense coding and error correction

Key distribution with BB84, QKD requirements and limitations, comparison with post-quantum cryptography, superdense coding and entropy. The final part describes quantum codes, syndromes and error recovery.

BB84 and key distribution
QKD: requirements, limitations and relationship with PQC
Superdense coding and entropy
Error correction and logical qubits

100 PDF pages · 0.6 MB

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Reading task

Compare quantum key distribution, superdense coding and error correction, specifying shared resources and what can be inferred from measurements.

Suggested preparation

Joint states, measurement bases and controlled circuits from the first three modules are useful preparation.

Topic map · 4

Material in Italian · original section titles

Destinations follow bookmarks in the original documents; the number refers to the physical PDF page, which may differ from the printed slide number.

Study → experiment

After reading the slides, you can open these activities to compare the module’s topics with the laboratory’s representations.

Opens in a new tab. Laboratory interface: Italian.

The slide numbering inside a PDF may differ from its total page count. Use the course index to move between modules.

← All courses

Quantum and Post-Quantum Computing.

Quantum and post-quantum

Course materials

Physical foundations and simulation of quantum systems

Reading task

Mathematical foundations: linear algebra, probability and measurement

Reading task

Quantum circuits: gates, reversibility and the Deutsch and Deutsch–Jozsa algorithms

Reading task

Quantum algorithms: Bernstein–Vazirani and Simon

Reading task

Shor’s algorithm: modular arithmetic, RSA and the Fourier transform

Reading task

Grover’s algorithm and amplitude amplification

Reading task

Quantum protocols: BB84, QKD, superdense coding and error correction

Reading task

Explore Topological Manifolds

Bloch sphere

Observe the figure

Figure key

Formula or construction

How to read the model

Lattice and basis

Observe the figure

Figure key

Formula or construction

How to read the model

Graph theory and cyber-attacks

From connections to attack paths

How this supports defence

Reading this figure

References

Explore the model

Interface preferences

Descending Fields

Scientific formulas

Topological
Manifolds

Opening the laboratory