2.7 Pauli operators

Adding to our collection of common single-qubit gates, we now look at the three Pauli operators36 \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.


  1. We use the standard basis \{|0\rangle,|1\rangle\} most of the time, and so often refer to operators as matrices.↩︎