J gate in measurement-based quantum computation ($J(\alpha )=H\mathrm{diag}(1,e^{i\alpha })$)

For the printed equatorial-basis convention,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.

The J gate is $J(\alpha )=HP(\alpha )$, where $P(\alpha )=\mathrm{diag}(1,e^{i\alpha })$: the phase gate acts first and the Hadamard gate acts second. Unitary invariance of the operator norm reduces its angle error to the single nonzero diagonal difference of the phase gate. Its norm is $|e^{i\delta }-1|=2|\sin (\delta /2)|\le |\delta |$. The quantum circuit gate-error telescoping bound then makes an angle tolerance of $ϵ/k$ sufficient for $k$ such imperfect gates and exact other gates.