Learning objectives
By the end of this lesson, you should be able to:
- Identify X, Y, and Z matrices.
- Predict their effect on basis states.
- Connect Pauli gates to π rotations of the Bloch sphere.
Prerequisites: What Is a Qubit?·Complex Numbers and Quantum Amplitudes
EXPERIMENT Interactive quantum laboratory
Change → Observe → ExplainManipulate the controls. Do not just watch the animation—predict what should happen before moving each parameter.
Three fundamental single-qubit gates
The Pauli matrices are
\[X=\begin{bmatrix}0&1\\1&0\end{bmatrix},\quad Y=\begin{bmatrix}0&-i\\i&0\end{bmatrix},\quad Z=\begin{bmatrix}1&0\\0&-1\end{bmatrix}.\]
The \(X\) gate swaps \(|0\rangle\) and \(|1\rangle\). The \(Z\) gate leaves \(|0\rangle\) unchanged and changes the sign of the \(|1\rangle\) amplitude. The \(Y\) gate combines a bit flip with a phase change.
Bloch-sphere interpretation
Up to global phase, the Pauli gates correspond to rotations by \(\pi\) around the x, y, and z axes of the Bloch sphere. This geometric interpretation becomes especially useful when introducing the continuous rotation gates \(R_x\), \(R_y\), and \(R_z\).
Key takeaways
What should stay with you
- X flips computational basis states.
- Z is a phase flip.
- Pauli gates are both matrices and geometric rotations.
Knowledge check
Which gate changes |1⟩ to −|1⟩ while leaving |0⟩ unchanged?
Z=diag(1,−1), so it changes the relative phase of the |1⟩ component.