QUANTUM MECHANICS AND ALGORITHMS

This book introduces quantum mechanics from an algebraic-geometrical point of view. This puts quantum mechanics quite firmly on top of three mathematical legs: linear algebra, calculus, and geometry. Thanks to the new geometrical picture, physics follows naturally from math. Not only that: quantum mechanics is well-embedded in science and technology. For example, the book uses quantum mechanics to develop quantum algorithms, including quantum FFT and Shor's factoring algorithm in cryptography. After introducing quantum mechanics from scratch, the discussion follows to entropy with a focus on entanglement entropy. Moreover, the algebraic study of angular momentum and energy helps analyse spin and polarization alike. Next, the book introduces algorithms in cryptography, where quantum algorithms are particularly relevant. Finally, Feynman diagrams are also introduced from an algebraic perspective. The book ends with examples of how practical algorithms are programmed on the computer.

Contents:

• Introduction to Quantum Mechanics:
  • Background: Machine Learning and Energy
  • Introduction to Quantum Mechanics from a Geometrical Point of View
  • Dynamics: Time–Energy Polar Coordinates
  • Toward Entanglement and The EPR Paradox
  • Application: Image Processing and Energy
• Entropy and Entanglement:
  • Entropy in Physics and Computing
  • Wave Function and the Fourier Transform
  • Entanglement Entropy
• Angular Momentum and Spin:
  • Energy and Angular Momentum: An Algebraic Study
  • Spin and Pauli Matrices
  • Spin and Polarization: An Algebraic-Geometrical Model
• Algorithms in Cryptography:
  • Coding–Decoding: The RSA Key Exchange
  • Fast Fourier Transform and Its Application
• Quantum Algorithms in Cryptography:
  • Quantum FFT: Recursive/Nonrecursive
  • Shor's Factoring Algorithm
• Feynman's Theory: An Algebraic Point of View:
  • Feynman Diagrams and The Commutator
  • Toward Quantum Field Theory
• Algorithms in C++:
  • Dynamic Tensors in C++
  • Maxwell's Equations: Linearize and Assemble on Finite Elements

Readership: This book is suitable for graduate and undergraduate students taking courses in Quantum mechanics (with a focus on its applications), and Cryptography with quantum computing.