9. Gates: Pauli Matrices
The Pauli Matrices
The Pauli Matrices are the alphabet of single-qubit quantum operations. Any single-qubit gate you ever see is a function or rotation based on these three matrices and the Identity matrix ($I$). They are the simplest, non-trivial, $2 \times 2$ Hermitian matrices (Postulate 3: they can act as observables) and they are unitary (Postulate 2: they are valid gates).
1. The Pauli Group ($\{I, X, Y, Z\}$)
| Gate | Matrix ($\sigma$) | Description |
|---|---|---|
| I | $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ | Identity: Do nothing. |
| $X$ ($\sigma_x$) | $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ | Bit-Flip (NOT): Flips $|0\rangle \leftrightarrow |1\rangle$. (Rotation about X-axis). |
| $Z$ ($\sigma_z$) | $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ | Phase-Flip: Flips the phase of $|1\rangle$. (Rotation about Z-axis). |
| $Y$ ($\sigma_y$) | $\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$ | Bit & Phase Flip: Combination of $X$ and $Z$. |
2. Physical and Mathematical Significance
- Observables: Since they are Hermitian ($M = M^\dagger$), they represent measurable physical properties (like spin).
- Rotations: On the Bloch sphere (Postulate 2), $X, Y, Z$ are the axes of rotation. Applying any Pauli matrix is a $180^\circ$ rotation about its respective axis.
- Eigenstates: The eigenstates of $X$ ($|+\rangle, |-\rangle$) and $Y$ ($|i+\rangle, |i-\rangle$) form alternative measurement bases.
3. The Relationship: Anti-Commutation
The Pauli matrices satisfy fundamental anti-commutation relations. For example:
$$XY = -YX$$This means the order in which you apply the gates absolutely matters. Changing the order of two operations in a quantum circuit changes the physics, which is often crucial for algorithms.
Your Task: Analyzing Pauli-Y
The Pauli-Y gate is the only one that explicitly contains the complex number $i$. This is where complex arithmetic is unavoidable. Apply the Pauli-Y gate to the computational basis state $|0\rangle$:
$$Y|0\rangle = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix}$$- What is the resulting column vector?
- Rewrite the result in Dirac notation (e.g., $\alpha|0\rangle + \beta|1\rangle$).
- Based on the final state, describe the physical effect of $Y|0\rangle$ in terms of bit flip and phase shift.