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

Conjugate transpose of a matrix

Since we now have the definition of the complex conjugate of a number, I'd like to quickly go over the conjugate transpose of a matrix as we will use this later in the book. The conjugate transpose is exactly as it sounds. It combines the notions of complex conjugates and the transposition of a matrix into one operation. If you remember from Chapter 2, The Matrix, we defined the transpose to be:

This is where we essentially convert the rows into columns and the columns into rows.

The conjugate of a matrix is just the conjugation of every entry:

For example, if the matrix M equals

,

then M* equals

.

So here is the big payoff. The conjugate transpose of a matrix A is defined to be:

The cross symbol at the top right of A is pronounced "dagger," and therefore when you hear "A dagger," the conjugate transpose of A is being referred to.

A quick example should get this all sorted. Let's use...