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

Cartesian form

We used the Cartesian form to define a complex number. To see why it is called Cartesian, notice we can also use an ordered pair of real numbers to represent the complex number z. The first number of the ordered pair will be the real part of the complex number, and the second number will be the imaginary part:

Given this, we can represent complex numbers on a Cartesian coordinate system since a and b are just real numbers. We will need to make a couple of modifications though.

We will replace the x axis with an axis for the real part of a complex number ( Re(z) ), and the y axis with an axis for the imaginary part of a complex number ( Im(z) ), like so:

Figure 6.1 – The complex plane

This is called the complex plane. Here is an example involving actual complex numbers:

Figure 6.2 – Complex numbers on the complex plane

Keep this in mind as we go through the basic operations of complex...