= J gate in measurement-based quantum computation
{c}
{title2=$J(\alpha)=H\operatorname{diag}(1,e^{i\alpha})$}
= J gate
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{synonym}
For the printed equatorial-basis convention,
$$
J(\alpha)=\frac1{\sqrt2}\begin{pmatrix}1&e^{i\alpha}\\1&-e^{i\alpha}\end{pmatrix}=H\operatorname{diag}(1,e^{i\alpha}).
$$
It is the <Hadamard gate> after a <phase gate>. The identities $J(\alpha)Z=XJ(\alpha)$ and $J(\alpha)X=e^{i\alpha}ZJ(-\alpha)$ follow by multiplying their matrices. For $\alpha=0,\pi/2$ these are <Clifford gates>. In particular $J(-\pi/2)=XJ(\pi/2)$, permitting a sign change of the measurement angle to be absorbed into a <Pauli frame>.
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