Principles of Superconducting Quantum Computers

2 Physics of Single Qubit Gates

2.1 Requirements for a Quantum Computer

As you might expect, having some way to represent and manipulate qubits is a basic requirement of any quantum computer. In thinking through all of the key requirements, DiVincenzo [3] identified the following five basic requirements:

1. A scalable physical system with well-characterized qubits
2. The ability to initialize the state of the qubits to a simple fiducial state, such as |000…⟩
3. Long relevant decoherence times, much longer than the gate operation time
4. A “universal” set of quantum gates
5. A qubit-specific measurement capability

In this Chapter we will discuss aspects of two of these requirements, but a bit out of DiVincenzo’s order:

• We will see that arbitrary manipulations of qubits can be interpreted as rotations, so we begin by discussing rotations—a subtopic of Requirement 4;
• We will then explore the physics of two-state systems needed to represent qubits—a subtopic of Requirement 1.

We will delay discussions of entanglement, measurement, initialization, and decoherence time to subsequent chapters.

2.2 Single Qubit Gates

2.2.1 Rotations

In the last chapter we encountered several types of gates and described them by $2\times 2$ matrices. To understand more generally what gates do to qubits we need to understand rotations. We will then see that the gates we have discussed can be viewed as particular rotations.

You may have encountered rotations in a classical physics or mechanics class, so we will begin with a brief ...