## Pauli operators

Adding to our collection of common single-qubit gates, we now look at the three **Pauli operators** \sigma_x, \sigma_y, and \sigma_z, also denoted by X, Y, and Z (respectively).
These three operators, combined with the identity, satisfy a lot of nice formal properties, which we shall examine briefly here, and then return to in more detail in Chapter 3.

**Identity** |
\mathbf{1}= \begin{bmatrix}1&0\\0&1\end{bmatrix} |
\begin{array}{lcr}|0\rangle&\longmapsto&|0\rangle\\|1\rangle&\longmapsto&|1\rangle\end{array} |

**Bit-flip** |
X = \begin{bmatrix}0&1\\1&0\end{bmatrix} |
\begin{array}{lcr}|0\rangle&\longmapsto&|1\rangle\\|1\rangle&\longmapsto&|0\rangle\end{array} |

**Bit-phase-flip** |
Y = \begin{bmatrix}0&-i\\i&0\end{bmatrix} |
\begin{array}{lcr}|0\rangle&\longmapsto&i|1\rangle\\|1\rangle&\longmapsto&-i|0\rangle\end{array} |

**Phase-flip** |
Z = \begin{bmatrix}1&0\\0&-1\end{bmatrix} |
\begin{array}{lcr}|0\rangle&\longmapsto&|0\rangle\\|1\rangle&\longmapsto&-|1\rangle\end{array} |

The identity is just a quantum wire, and we have already seen the X and Z gates in Phase gates galore, as the bit-flip and phase-flip (respectively).
Note that, of these latter two, only the X gate has a classical analogue (as the logical \texttt{NOT} operator).
The remaining gate, the Y operator, describes the combined effect of both the bit- and the phase-flip: ZX=iY.

In fact, this is just one of the equations that the Pauli matrices satisfy.
The Pauli matrices are unitary and Hermitian, they square to the identity, and they anti-commute.
By this last point, we mean that
\begin{aligned}
XY+YX&=0,
\\XZ+ZX&=0,
\\YZ+ZY&=0.
\end{aligned}
As already mentioned, they satisfy ZX=iY, but also any cyclic permutation of this equation.

These operators are also called **sigma matrices**, or **Pauli spin matrices**.
They are so ubiquitous in quantum physics that they should certainly be memorised.