# What are the “other” Hadamard matrices?

The Pauli matrices $$X = \begin{pmatrix}0&1\\1&0\end{pmatrix}, Y=\begin{pmatrix}0&-i\\i &0\end{pmatrix},\,\text{and}\, Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$$ can be used to construct the Hadamard gate $$H=\frac{1}{\sqrt{2}}(X+Z)=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\1&-1\end{pmatrix}.$$ $H$ is Hermitian and two other Hermitian matrices arise when doing a similar computation with the $Y$ matrix: $$\frac{1}{\sqrt{2}}(X+Y)=\frac{1}{\sqrt{2}}\begin{pmatrix}0 & 1-i \\ 1+i & 0\end{pmatrix}\quad\text{and}\quad\frac{1}{\sqrt{2}}(Y+Z)\begin{pmatrix}1 & -i \\ i & -1\end{pmatrix}$$ Do these matrices have some names and if yes, do there exist known properties about them?

• Are you curious about the relation between Hadamard gate and Pauli matrices? There may be no special names related to Hadamard for the new matrices you constructed. – Eden Harder Oct 8 '14 at 1:47
• @EdenHarder: I am investigating the relation between these 3 matrices and the Pauli matrices for quite some time now. I was just wondering whether these matrices have appeared explicitly somewhere in the literature. – Mathias Soeken Oct 8 '14 at 9:23

Realize that an arbitrary rotation around the axis $\mathbf{n}$ is given by $R_\mathbf{n}=\cos(\alpha/2)I-i\sin(\alpha/2)\hat{\mathbf{n}}\cdot\mathbf{\sigma}$ and an arbitrary unitary operator can be written as $U=\exp{(i\gamma)}R_\mathbf{n}$ with $\gamma$ some phase factor. Thus, in general, any operations on the qubit can be seen as a rotation with some phase factor. Nevertheless, in the literature only the Hadamard, X,Y,Z and the phase shift gate are usually mentioned since these gates are conceptually the most practical to use on a single qubit.