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

The inner product

An inner product can actually be any function that follows a few properties, but we are going to zero in on one definition of the inner product that we will use in quantum computing. Here it is:

Mathematicians use the preceding notation for the inner product, but Dirac defined it with a bra and ket, calling it a bracket:

Now, if we define a bra to be the conjugate transpose of its corresponding ket, so that if:

Then, x| is now:

We can then define a bracket as just matrix multiplication!

Pretty cool, eh? That is one of the reasons why bra-ket notation is so convenient! You should notice something else too. The bra x| is a linear functional. It will take any vector |y and give you a scalar according to the inner product formula!

Let's look at an example. Let's say |x and |y are defined this way:

...