= Pauli-frame propagation along a measurement wire
{c}
{title2=$U(\eta)X^pZ^q\simeq X^qZ^pU((-1)^p\eta)$}
With <equatorial qubit measurement> vectors $(|0\rangle+(-1)^s e^{i\eta}|1\rangle)/\sqrt2$, the gate in <one-bit teleportation> is $U(\eta)=H\operatorname{diag}(1,e^{-i\eta})=J(-\eta)$. Direct multiplication gives $U(\eta)Z=XU(\eta)$ and $U(\eta)X=e^{-i\eta}ZU(-\eta)$. Moving a <Pauli frame> through this gate therefore exchanges its $X$ and $Z$ exponents and reverses the angle if the old $X$ exponent is one, as displayed up to <global phase>. A measurement outcome $s$ adds a further $X^s$ factor. This calculation supplies adaptive angle choices and the final classical output-bit correction of a linear <cluster state> computation.
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