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

Orthonormality

In this section, we will look at the concepts of the norm and orthogonality to come up with orthonormality.

The norm

We can define a metric on our vector spaces called the norm and denote it this way, x, where x is the vector on which the norm is being measured. In two- and three-dimensional Euclidean spaces, it is often called the length of a vector, but in higher dimensions, we use the term norm. It gives us a way to measure vectors.

We define the norm using our inner product from the previous section, like so:

As always, let's look at an example. What is the norm of the vector |xhere?

Well, let's work it out:

As you can see, the norm, x, of |x is the square root of 29.

Normalization and unit vectors

Oftentimes, especially in quantum computing, we will want to represent our vectors by something called a unit vector. The word unit refers...