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

Operators

In this section, we will consider linear operators. We first described these in Chapter 5, Transforming Space with Matrices. To reiterate, linear operators are linear transformations that map vectors from and to the same vector space. They are represented by square matrices. For just this section, I will put a "hat" or caret on the top of operators and use just the uppercase letter for matrices, as I want to be deliberate when referencing operators.

For instance, let's look at the operator that transforms the zero and one states:

Now, let's come up with a matrix that represents this operator. The question becomes, which basis will we use? Let's use the computational basis, which is |0 and |1⟩. I will denote this set of basis vectors by the letter C. So, the operator in the C basis is represented by:

Now, I want to come up with a matrix representation of in the |+, |- basis...