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 5: Using Matrices to Transform Space

Linear transformations are one of the most central topics in linear algebra. Now that we have defined vectors and vector spaces, we need to be able to do things with them. In Chapter 3, Foundations, we manipulated mathematical objects with functions. When we manipulate vectors in vector spaces, mathematicians use the term linear transformations. Why the change of terminology? As with most things in linear algebra, the wording is inspired by Euclidean geometry. We will see that, geometrically, these "functions" actually "transform" vectors from one direction and length to another. But this visual transformation has been generalized algebraically to all types of vectors (n-tuples of numbers, functions, and so on).

We also go through the crucial link between linear transformations and matrices. The most important point of this chapter is that linear transformations can always be represented by matrices when the vector...