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For $|v_s(\eta )⟩=(|0⟩+(-1{)}^s{e}^{i\eta }|1⟩)/\sqrt{2}$, adding $\pi$ exchanges the two outcomes of the same projective measurement. At an angle $\varphi =j\pi /2$, the choices $\varphi$ and $-\varphi$ therefore specify the same unordered pair of projectors: their labels differ by $j\mathrm{mod}2$. An adaptive choice between these signs can be replaced by a fixed measurement and a classical label exchange. Together with Pauli-frame propagation along a measurement wire, this permits simultaneous measurements when all adaptive sign changes occur at these Clifford gate angles.

Using the printed measurement-vector convention, the unitary operator isThus it is a phase gate followed by a Hadamard gate, with angle sign opposite to the parameter in the catalogued J gate. Its rows are the conjugate transposes of the orthonormal measurement vectors, so it is unitary. Right multiplication by the Pauli Z gate changes the sign of the second column, which is exactly the effect of left multiplication by the Pauli X gate, exchanging the two rows. Explicitly,Similarly, right multiplication by the Pauli X gate exchanges columns, givingTherefore the exact identities, retaining the global phase in the second one, areThese supply Pauli-frame propagation along a measurement wire.

Label the chain cluster state from top to bottom by $1,2,3,4$, and let $k,l,m$ be the phase-measurement outcomes on the first three vertices, with raw computational basis outcome $s$ on vertex four. Repeated one-bit teleportation leaves the last qubit, up to global phase, inThis sequential description applies even though the whole graph state is prepared first: a later Controlled-Z gate acts only on unmeasured qubits and commutes with earlier single-qubit measurement operators. It can therefore be moved past those measurements for the purpose of calculating each conditional branch.

Take $\alpha =\theta$ and $\beta =0$, so $U(0)=H$. After the second measurement the logical state isFor binary $p,q$, the identities proved above imply the Pauli-frame propagation along a measurement wire rulewhere $\simeq$ means equality up to global phase. Choose the third angle adaptively as $\gamma =(-1{)}^{l\oplus 1}\varphi$. Then $(-1{)}^l\gamma =-\varphi$, andReading the target circuit in temporal order gives its premeasurement stateConsequentlyThe Pauli Z gate changes only phases in the computational basis, while the Pauli X gate exchanges its two measurement outcomes. Thus the explicit choices areIndeed, for every earlier outcome branch,Hence averaging over the earlier random outcomes leaves exactly the target probability distribution. This implements the desired measurement-based quantum computation without physically removing the final Pauli frame.