This talk explores the mathematical principles behind quantum algorithms and their use in numerical computation. We will discuss how vectors and matrices are represented on a quantum computer, how computations are carried out, and what information can be extracted through measurement. Using linear systems and partial differential equations as examples, we will see how classical ideas such as discretization and preconditioning enter the design and analysis of quantum algorithms. Complexity estimates and numerical experiments illustrate both their potential computational advantages and the challenges of their practical realization.