The Exercise

A single qubit has the computational basis states $|0\rangle$ and $|1\rangle$, and in that basis its operators are two by two complex matrices. A unitary $U$ moves an operator $A$ to $U A U^\dagger$. The exercise takes the phase gate $S$ for $U$ and asks what this conjugation makes of the Pauli matrices $X$ and $Z$.

The Pauli Matrices and the Phase Gate

The Pauli matrices $X$, $Y$ and $Z$ of a qubit are the two by two complex matrices, with rows and columns labelled by the basis states $|0\rangle$ and $|1\rangle$, $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$(1)

The phase gate $S$ is the diagonal two by two complex matrix that leaves $|0\rangle$ unchanged and multiplies $|1\rangle$ by the imaginary unit $i$, $$S = \operatorname{diag}(1, i) = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix}.$$(2)^✦ Its adjoint $S^\dagger$ is the conjugate transpose, $\operatorname{diag}(1, -i)$.

The Claim

What Is Checked

The claim is stated over the four matrices $X$, $Y$, $Z$ and $S$, each given by its entries in (1) and (2), with $S^\dagger$ the conjugate transpose of $S$.

About these matrices the library states the two identities (3) and (4) as equalities of two by two complex matrices.