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

Tensor products

Tensor products are a way to combine vector spaces. One of the postulates of quantum mechanics is that the state of a qubit is completely described by a unit vector in a Hilbert space. The problem then becomes how to deal with more than one qubit. This is where a tensor product comes in. Each qubit has its own Hilbert space, and to describe many qubits as a system, we need to combine all their Hilbert spaces into one bigger Hilbert space.

Mathematically, that means that if we have a Hilbert space H and another Hilbert space J, we denote their tensor product as:

If H is an h dimensional space and J is a j dimensional space, then the dimension of the combined space M is h j. In other words:

Before we go any farther, let's look at the tensor product of two vectors.

The tensor product of vectors

The tensor product of two vectors is denoted in the following way in bra-ket notation. You'll notice that there are four different ways...