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

Special types of matrices

In the world of matrices, some are so special that they have been singled out. Here they are.

Square matrices

A special type of matrix is a square matrix. A square matrix is one where the number of rows equals the number of columns. In other words, it is an m × n matrix in which m = n. Square matrices show up all over the place in quantum computing due to special properties that they can have—for example, symmetry, which is discussed later in the book. As we progress in the book, they will become one of the central types of matrices we will use. Some examples of square matrices are:

Identity matrices

An important type of square matrix is an identity matrix, named I. It is defined so that it acts as the number 1 in matrix multiplication so that the following holds true:

It has ones all down its principal diagonal and zeros everywhere else. Its dimensions need to change based on the matrix it is being multiplied by. Here are...