Discrete and Quantum Fourier Transform - Beyond Shor's Algorithm - Part 1

Learn about the fundamentals and advanced applications of Quantum Fourier Transform (QFT) in this comprehensive lecture from András Gilyén at the Park City Mathematics Institute. Explore the mathematical foundations of discrete and quantum Fourier transforms, moving beyond their traditional use in Shor's algorithm and hidden subgroup problems. Delve into Jordan's gradient computation algorithm and its applications in convex optimization, distribution learning, and quantum tomography. Master recent improvements in phase estimation, including bias removal and coherent boosting techniques. Examine QFT from the Heisenberg picture perspective and understand operator Fourier transform's relevance to quantum computing. Through detailed explanations and accompanying problem sets, gain practical experience with concepts like finite group transformations, Abelian Hidden Subgroup Problems, and measurement outcomes in quantum states. Part of a broader quantum computation program featuring leading experts, this lecture serves as an essential foundation for understanding modern quantum computing applications and their mathematical underpinnings.