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

Cauchy-Schwarz and triangle inequalities

The Cauchy-Schwarz inequality is one of the most important inequalities in mathematics. Succinctly stated, it says that the absolute value of the inner product of two vectors is less than or equal to the norm of those two vectors multiplied together. In fact, they are only equal if the two vectors are linearly dependent:

There are several proofs of this inequality, which I encourage you to seek out if you are interested. But, in the totality of things, knowing this inequality is all that is really required for quantum computing.

The other major inequality is the triangle inequality. It comes from our old friend Euclid in his book The Elements. Succinctly stated, it says that the length of two sides of a triangle must always be more than the length of one side. They will only be equal in the corner case when the triangle has zero area. It is very intuitive once you see some example triangles. Here are some triangles...