2.7 Pauli operators

Adding to our collection of common single-qubit gates, we now look at the three Pauli operators55 \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. After that, these operators will turn up everywhere, so it’s good to get familiar with them!

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 anticommute. 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.

  1. Most of the time we refer to “operators” as “matrices”, where the implicit assumption is that we are using the standard basis \{|0\rangle,|1\rangle\}.↩︎