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

Spectral decomposition

The spectrum of a square matrix is the set of its eigenvalues. There is a cool theorem in linear algebra that states that all matrices representing a linear operator have the same spectrum. Before we use the spectrum though, we need to talk about diagonal matrices.

Diagonal matrices

The main diagonal of a matrix is every entry where the row index equals the column index. Examples make this very easy to see. All the following matrices have the letter d on their main diagonal:

Now, a diagonal matrix has zero on all entries outside the main diagonal. Here are examples of diagonal matrices:

Here are two cool features of diagonal matrices that make them all the rage at linear algebra parties. One, all their eigenvalues are on their main diagonal. Two, they are very easy to exponentiate. Let's see the latter in action real quick:

Try to exponentiate any regular old random matrix...