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

Singular value decomposition

Singular Value Decomposition (SVD) is probably the most famous decomposition you can do for linear operators and matrices. It is at the core of search engines and machine learning algorithms. Additionally, it can be used on any type of matrix, even rectangular ones. However, we will only look at square matrices.

Succinctly stated, it guarantees that for any matrix A, it can be decomposed into three matrices:

Whereas U is a unitary matrix, Σ (sigma) is a diagonal matrix with what is known as the singular values of A on its diagonal, and V is also a unitary matrix. It should be noted that this decomposition is not unique, and different matrices can be used for U, Σ, and V.

Let's look at an example. We have the following matrix A:

Without going through the math, I'm going to tell you that SVD can be used to get this decomposition:

Let's make sure...