Lecture Notes on Quantum Computation

Link to the lecture notes

This set of notes encapsulates most of the material I am teaching on the topic of quantum algorithm, quantum programming and quantum compiling. The target is a student at Master level in Computer Science, with some acquaintance with mathematics. Section 2 attempts at covering the mathematical knowledge necessary to follow.

The document is available as a PDF file. It is regularly updated: the version number corresponds to the last date of edition. The current version is v.2026.02.07.

There are (probably) many typos hiding in the document! Do not hesitate to send me the ones you find so that I can correct them in a later version.

Table of Contents

  1. Introduction
  2. Mathematical Background
    1. Notations
    2. Sums and series
    3. Complex Numbers
    4. Vector Space
    5. Scalar Product
    6. Kets and Bras
    7. Kronecker product
    8. Linear Maps
    9. Hermitian and Unitary Maps
    10. Exercices
  3. Qubit-based Computation
    1. The Quantum Co-Processor Model
    2. One Quantum Bit
    3. Several Quantum Bits
    4. The Quantum Circuit Model
    5. Quantum Gates on 1 Qubit
    6. Quantum Gates on Several Qubits
    7. Creating New Quantum Registers
    8. Reading Quantum Registers
    9. Discarding Quantum Registers
    10. Exercices
  4. Hardware Constraints and Circuit Synthesis
    1. A bit of Complexity Theory
    2. Low-level gate-sets
    3. Universality of CNOT and 1-qubit rotations
    4. Tradeoffs: a Case-Study
    5. Quantum Computation with Magic States
    6. Measurement-Based Quantum Computation
    7. Classical Computation in the Co-Processor
    8. A Word on Hardware
    9. Focus: Linear Optical Quantum Computation
    10. Exercises
  5. Structure of Quantum Algorithms
    1. High-Level View
    2. Oracles
    3. Encoding Natural Numbers
    4. Amplitude Amplification
    5. Quantum Fourier Transform
    6. Phase Estimation
    7. Trotterization
    8. Exercises
  6. Algorithms for LSQ era
    1. Simple Oracle-Based Algorithms
    2. Shor
    3. HHL
    4. Exercises
  7. Algorithms for NISQ era
    1. Variational Algorithms
    2. VQE
    3. QAOA
    4. Example: MAXCUT and QUBO problems
    5. Exercises

History

v.2026.02.07
Reformulating Sec 5.4.1.
v.2026.02.01
Sec 7.3 split in 7.3 and 7.4, introduction of the notion of QUBO.