Special Quantum Operations: Pauli Matrices

Pauli Matrices as Quantum Gates

There is a set of quantum operations that is used very frequently in quantum theory.  

These matrices are known as the Pauli matrices.

They are:  

$$\begin{align*} \sigma_x = X &= \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \\ \sigma_y = Y &= \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}, \\ \sigma_z = Z &= \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}. \end{align*}$$

Sometimes, the identity matrix $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ is also included and given the label $\sigma_0$.

They are important to memorize because they come up so often in quantum computing. They are some of the most important quantum gates i.e. quantum operations.

This terminology comes from computer science, where operations like AND, OR, and XOR are called "gates". Don't worry if you don't know what those are, you can simply think of them as the "building blocks" of classical computers (i.e. the computers we're used to from everyday life).

You can think of the Pauli matrices roughly as a quantum equivalent of these classical gates, hence why we call them quantum gates. We will soon encounter two qubit and multi qubit gates, which are indeed much closer to the classical analogues, later.

They form the building blocks of some of the most important quantum algorithms. 

Remembering where the minus sign goes

It is very common for people who are first learning about the Pauli matrices to forget where the minus sign goes in the $\sigma_{y}$ and $\sigma_{z}$ matrices. Here is a trick to help remember: the minus sign is always closest to the visual center of the matrix.

For $\sigma_{y}$ for example, the minus sign being in the top right entry makes sense, because that puts it closest to the visual center of the matrix (as opposed to the bottom left entry which would put the minus sign far from the center). 

Descriptions of the actions of the Paulis

$\sigma_x$ is sometimes called the bit-flip operator because:

$$\begin{align*} \sigma_x |0\rangle &= \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \end{bmatrix} = |1\rangle, \\ \, \\ \sigma_x |1\rangle &= \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} = |0\rangle, \end{align*}$$

i.e. it "flips" the bits $|0\rangle$ and $|1\rangle$ when acting on them.  

Similarly, $\sigma_z$ is sometimes called the phase-flip operator:  

\begin{align*} ...