Chapter 1
Foundations of Quantum Computing
The beginning is always today.
— Mary Shelley
You may have heard that the mathematics needed to understand quantum computing is arcane, mysterious and difficult…but we utterly disagree! In fact, in this chapter, we will introduce all the concepts that you will need in order to follow the quantum algorithms that we will be studying in the rest of the book. Actually, you may be surprised to see that we will only rely on some linear algebra and a bit of (extremely simple) trigonometry.
We shall start by giving a quick overview of what quantum computing is, what the current state of the art is, and what the main applications are expected to be. After that, we will introduce the model of quantum circuits. There are several computational models for quantum computing, but this is the most popular one and, moreover, it's the one that we will be using throughout most of the book. Then, we will describe in detail what qubits are, how we can operate on them by using quantum gates, and how we can retrieve results by performing measurements. We will start with the simplest possible case — just a humble qubit! Then, we will steadily build upon that until we learn how to work with as many qubits as we want.
This chapter will cover the following topics:
Quantum computing: the big picture
The basics of the quantum circuit model
Working with one qubit and the Bloch sphere
Working with two qubits and entanglement
Working with multiple qubits and universality
After reading this chapter, you will have acquired a solid understanding of the fundamentals of quantum computing and you will be more than ready to learn how practical quantum algorithms are developed.
