Pauli-frame propagation along a measurement wire ($U(\eta )X^p{Z}^q\simeq X^q{Z}^{p}U((-1{)}^p\eta )$)

With equatorial qubit measurement vectors $(|0⟩+(-1{)}^s{e}^{i\eta }|1⟩)/\sqrt{2}$, the gate in one-bit teleportation is $U(\eta )=H\mathrm{diag}(1,e^{-i\eta })=J(-\eta )$. 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.