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

Basis

The word basis is used often in English speech and its colloquial definition is actually a good way to look at the word basis in linear algebra:

Basis, ba·sis \ ˈbā-səs \ plural bases\ ˈbā-ˌsēz \ Noun

Something on which something else is established or based. Example 1: Stories with little basis in reality. Example 2: No legal basis for a new trial.

The reason for this is that you can choose different bases for a vector space. While the vector space itself does not change when you choose a different basis, the way things are described with numbers does.

Let's look at an example in 2. Consider the vector |u, given as follows:

Figure 4.11 – Graph of the vector |u

Clearly, its coordinates are (3,3). What if I told you I could describe the same vector with the coordinates (3,0)? Wait a minute; that should disturb you. We never talk about the basis in most math...