Introduction to the Quantum Fourier Transform, Phase Estimation, and Linear Algebra Techniques for Quantum Electromagnetic Solvers

Quantum algorithms can reduce the polynomial cost of the best-known classical methods to logarithmic scaling for many fundamental problems, including the Fourier transform and the solution of large matrix equations. This presentation introduces the basic principles of quantum information, showing how data is encoded in qubits and manipulated through standard quantum gates to form complex circuits capable of solving applied problems in microwave engineering and computational electromagnetics. We will first explain the mechanics of the Quantum Fourier Transform (QFT) and Quantum Phase Estimation (QPE) as standalone algorithms, then illustrate their role in constructing the Harrow–Hassidim–Lloyd (HHL) algorithm for solving linear systems. Finally, we will discuss recent advances in tensor networks and their use for compressing and processing large datasets. In particular, we will highlight how classical matrices arising from the Method of Moments and the Finite Element Method can be efficiently mapped into quantum states, where the number of qubits required scales only logarithmically with the matrix dimension. This perspective underscores the potential of combining quantum algorithms with tensor-network techniques to address exa-scale problems in computational electromagnetics.