## 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 later on, in Section 3.3.

**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 (Section 2.6) the X and Z gates as the bit-flip and phase-flip, respectively.
Note that, of the X and Z gates, only the X gate has a classical analogue (namely 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,
\\XZ&=-ZX,
\\YZ&=-ZY.
\end{aligned}
As already mentioned, they satisfy ZX=iY, but also any cyclic permutation of this equation (that is, replace X with Y, Y with Z, and Z with X, and repeat this as many times as you wish).

These operators are also called **sigma operators** (usually when we use the notation \sigma_x, \sigma_y, \sigma_z) or (when written as matrices in the standard basis, as we have done) as **Pauli spin matrices**.
They are so ubiquitous in quantum physics that they should certainly be memorised.

### From bit-flips to phase-flips, and back again

The Pauli Z gate is a special case of a phase gate P_\varphi with \varphi=\pi.
When we insert it into the interference circuit we obtain

If you wish to verify this, write the Hadamard gate as H = (X+Z)/\sqrt{2} and use the properties of the Pauli operators.
So the Hadamard gate turns phase-flips into bit-flips, but it also turns bit-flips into phase-flips:

Let us also add, for completeness, that HYH=-Y.
You will see these identities again and again, especially when we discuss quantum error corrections.
\begin{aligned}
HXH &= Z
\\HZH &= X
\\HYH &= -Y
\end{aligned}