2.7 Pauli operators

Adding to our collection of common single-qubit gates, we now look at the three Pauli operators48 \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.

2.7.1 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.49 \begin{aligned} HXH &= Z \\HZH &= X \\HYH &= -Y \end{aligned}

  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\}.↩︎

  2. Unitaries, such as H, that take the three Pauli operators to the Pauli operators via conjugation form the so-called Clifford group, which we will meet later on. Which phase gate is in the Clifford group of a single qubit?↩︎