Book Image

Essential Mathematics for Quantum Computing

By : Leonard S. Woody III
5 (1)
Book Image

Essential Mathematics for Quantum Computing

5 (1)
By: Leonard S. Woody III

Overview of this book

Quantum computing is an exciting subject that offers hope to solve the world’s most complex problems at a quicker pace. It is being used quite widely in different spheres of technology, including cybersecurity, finance, and many more, but its concepts, such as superposition, are often misunderstood because engineers may not know the math to understand them. This book will teach the requisite math concepts in an intuitive way and connect them to principles in quantum computing. Starting with the most basic of concepts, 2D vectors that are just line segments in space, you'll move on to tackle matrix multiplication using an instinctive method. Linearity is the major theme throughout the book and since quantum mechanics is a linear theory, you'll see how they go hand in hand. As you advance, you'll understand intrinsically what a vector is and how to transform vectors with matrices and operators. You'll also see how complex numbers make their voices heard and understand the probability behind it all. It’s all here, in writing you can understand. This is not a stuffy math book with definitions, axioms, theorems, and so on. This book meets you where you’re at and guides you to where you need to be for quantum computing. Already know some of this stuff? No problem! The book is componentized, so you can learn just the parts you want. And with tons of exercises and their answers, you'll get all the practice you need.
Table of Contents (20 chapters)
1
Section 1: Introduction
4
Section 2: Elementary Linear Algebra
8
Section 3: Adding Complexity
13
Section 4: Appendices
Appendix 1: Bra–ket Notation
Appendix 2: Sigma Notation
Appendix 5: References

Chapter 7: EigenStuff

Eigen (pronounced EYE-GUN) is a German prefix to words such as eigentum (property), eigenschaft (a feature or characteristic), and eigensinn (an idiosyncrasy). To sum up, we are looking for some values and vectors that are characteristic, idiosyncratic, and a property of something. What is that something? That something is our old friend the linear operator and its representations as square matrices. But before we get there, we'll need to look at some other concepts such as the matrix inverse and determinant. We'll wrap it all up with the trace of a matrix and some properties that the trace, determinant, and eigenvalues all share. These concepts will allow us to reach even further heights in the chapters that follow.

In this chapter, we are going to cover the following main topics:

  • The inverse of a matrix
  • Determinants
  • Invertible matrix theorem
  • Eigenvalues and eigenvectors
  • Trace
  • The special properties of eigenvalues
...