Up to this point, we have treated the qubit as a static vector within a Hilbert space. To perform computation, we must manipulate this state. Just as classical computers use logic gates to process bits, quantum systems use unitary operators to transform qubits. This chapter focuses on these operations, known as quantum gates, and how they alter the probability amplitudes of a state vector.

We begin by establishing the mathematical definition of superposition. Unlike a classical bit that must be in a definite state, a qubit can exist as a linear combination of the standard basis vectors $|0\rangle$ and $|1\rangle$. You will learn to represent this state using the equation:

$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$

The material then covers the standard library of single-qubit operations. We examine the Pauli gates ($X$, $Y$, and $Z$), which correspond to rotations around the axes of the Bloch sphere. Following this, we introduce the Hadamard gate, the primary operator used to generate superposition from a basis state.

We conclude by addressing the mechanics of measurement. You will see how observing a quantum system forces the state vector to collapse into a single outcome. To reinforce the theory, you will write Python code to initialize qubits, apply gate operations, and simulate measurement results in a local environment.