Appears in collection : CEMRACS 2025: Quantum Computing / CEMRACS 2025: Calcul quantique

Quantum computing promises to transform computational capabilities across diverse fields. The rapid advancement of quantum algorithms has expanded the potential of quantum computing for tackling a broad spectrum of scientific computing challenges. In this lecture, we will present fundamental concepts of quantum algorithms, focusing on solving large-scale numerical linear algebra problems as well as addressing high-dimensional linear and nonlinear differential equations. We will start with basic notions of quantum states, unitary operators, no-cloning theorem and measurements. After introducing block-encoding and linear combination of unitaries (LCU), we will discuss various quantum algorithms for scientific computing, i.e., Quantum Linear System Problem (QLSP), Quantum Singular Value (Eigenvalue) Transformation (QSVT), Hamiltonian Simulation and Trotterization, Adiabatic Quantum Computation (AQC), Variational Quantum Eigensolver (VQE), Quantum Krylov Algorithms and Quantum (linear) Differential Equation Solvers.