Essential Mathematics for Quantum Computing

Leonard S. Woody III

• Publisher: Packt Publishing
• Year: 2022
• ISBN:
  9781801070188
• Edition: 1
• Topic: Matematica
• Subtopic: Algebra lineare

Demystify quantum computing by learning the math it is built onKey Features• Build a solid mathematical foundation to get started with developing powerful quantum solutions• Understand linear algebra, calculus, matrices, complex numbers, vector spaces, and other concepts essential for quantum computing• Learn the math needed to understand how quantum algorithms functionBook DescriptionQuantum 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.What you will learn• Operate on vectors (qubits) with matrices (gates)• Define linear combinations and linear independence• Understand vector spaces and their basis sets• Rotate, reflect, and project vectors with matrices• Realize the connection between complex numbers and the Bloch sphere• Determine whether a matrix is invertible and find its eigenvalues• Probabilistically determine the measurement of a qubit• Tie it all together with bra-ket notationWho this book is forIf you want to learn quantum computing but are unsure of the math involved, this book is for you. If you've taken high school math, you'll easily understand the topics covered. And even if you haven't, the book will give you a refresher on topics such as trigonometry, matrices, and vectors. This book will help you gain the confidence to fully understand quantum computation without losing you in the process!

Section 1: Introduction

This section starts the book off with easy concepts such as vectors and matrices.

The following chapters are included in this section:

• Chapter 1, Superposition with Euclid
• Chapter 2, The Matrix

Chapter 1: Superposition with Euclid

Mathematics is the language of physics and the foundation of computer science. Since quantum computing evolved from these two disciplines, it is essential to understand the mathematics behind it. The math you need is linear in nature, and that is where we will start. By the time we are done, you will have the mathematical foundation to fundamentally understand quantum computing. Let's get started!

In this chapter, we are going to cover the following main topics:

• Vectors
• Linear combinations
• Superposition

Vectors

A long time ago in a country far, far away, there lived an ancient Greek mathematician named Euclid. He wrote a book that defined space using only three dimensions. We will use his vector space to define superposition in quantum computing. Don't be fooled—vector spaces have evolved tremendously since Euclid's days, and our definition of them will evolve too as the book progresses. But for now, we will stick to real numbers, and we'll actually only need two out of the three dimensions Euclid proposed.

To start, we will define a Euclidean vector as being a line segment with a length or magnitude and pointing in a certain direction, as shown in the following screenshot:

Figure 1.1 – Euclidean vector

Two vectors are equal if they have the same length and direction, so the following vectors are all equal:

Figure 1.2 – Equal vectors

Vectors can be represented algebraically by their components. The simplest way to do this is to have them start at the origin (the point (0,0)) and use their x and y coordinates, as shown in the following screenshot:

Figure 1.3 – Vectors represented geometrically and algebraically

You should note that I am using a special notation to label the vectors. It is called bra-ket notation. The appendix has more information on this notation, but for now, we will use a vertical bar or pipe, |, followed by the variable name for the vector and then an angle bracket, ⟩, to denote a vector (for example, |a⟩). The coordinates of our vectors will be enclosed in brackets [ ]. The x coordinate will be on top and the y coordinate on the bottom. Vectors are also called "kets" in this notation—for example, ket a, but for now, we will stick with the name vector.

Vector addition

So, it ends up that we can add vectors together both geometrically and algebraically, as shown in the following screenshot:

Figure 1.4 – Vector addition

As you can see, we can take vectors and move them in the XY-plane as long as we preserve their length and direction. We have taken the vector |b⟩ from our first graph and moved its start position to the end of vector |a⟩. Once we do that, we can draw a ...