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

Vector space

Now that we have covered all of the abstract concepts we need to understand, we can give a formal definition of a vector space, before looking at the implications of these in the following chapters.

A vector space is defined as having the following mathematical objects:

  1. An Abelian group V,+ with an identity element e. We call members of the set V vectors. We define the identity element to be the zero vector, and we denote this by 0. The operation + is called vector addition.
  2. A field {F, +, }. We say that V is a vector space over the field F, and we call the members of F scalars.

    The Zero Vector Is Not Denoted by |0

    It is important to note that we denote the zero vector with a bold zero – 0 – and it is totally different from the vector |0 we defined earlier in the book. This is the convention in quantum computing.

We can define an additional operation as scalar multiplication, which is an operation...