Now that we have the mathematical and physical preliminaries under our belt, we can move on to the nuts and bolts of quantum computing. At the heart of a classical computer is the notion of a bit and at the heart of quantum computer is a generalization of the concept of a bit called a qubit, which shall be discussed in Section 5.1. In Section 5.2, classical (logical) gates, which manipulate bits, are presented from a new and different perspective. From this angle, it is easy to formulate the notion of quantum gates, which manipulate qubits. As mentioned in Chapters 3 and 4, the evolution of a quantum system is reversible, i.e., manipulations that can be done must also be able to be undone. This “undoing” translates into reversible gates, which are discussed in Section 5.3. We move on to quantum gates in Section 5.4.

BITS AND QUBITS

What is a bit?

Definition 5.1.1Abitis a unit of information describing a two-dimensional classical system.

There are many examples of bits:

1. ▪ A bit is electricity traveling through a circuit or not (or high and low).

2. ▪ A bit is a way of denoting “true” or “false.”

3. […]

Quantum theory is cast in the language of complex vector spaces. These are mathematical structures that are based on complex numbers. We learned all that we need about such numbers in Chapter 1. Armed with this knowledge, we can now tackle complex vector spaces themselves.

Section 2.1 goes through the main example of a (finite-dimensional) complex vector space at tutorial pace. Section 2.2 provides formal definitions, basic properties, and more examples. Each of Section 2.3 through Section 2.7 discusses an advanced topic.