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Let $1$ be a leaf with neighbour $2$ and let the graph on the other vertices be generated by the Controlled-Z gate product $E_{\mathrm{rest}}$. If the leaf has Pauli Z gate frame $Z_1^r$, measuring it at angle $\alpha$ with result $s$ applies one-bit teleportation and leavesAny known Pauli Z gate frames on other vertices remain as additional factors before those vertices' input states. The identity $J(\alpha )Z^r=X^{r}J(\alpha )$ combines the leaf byproduct with the measurement result.

Label a four-cycle $0-1-2-3-0$. Measure vertex $0$ in the computational basis with result $r$, then vertex $1$ in the equatorial qubit measurement basis at angle $\alpha$ with result $s$. Graph-state vertex deletion and one-bit teleportation leaveFinal computational basis measurements with raw results $d_2,d_3$ therefore simulate the ideal circuit $E_{23}J(\alpha {)}_2$ on $|++⟩$ after the classical correction $k=d_2\oplus r\oplus s$, $l=d_3$. The output probability distribution is correct in every prior branch; no physical Pauli frame correction is needed.

Two angle-zero one-bit teleportations with outcomes $p,q$ implement $X^{q}H{X}^{p}H=X^q{Z}^p$ on an arbitrary input, up to global phase. A Controlled-Z gate with a fixed $|1⟩$ quantum ancilla implements $Z$, so $Z^p$ removes the known $Z$ factor. The result is $X^q$ up to phase. With probability one half the desired Pauli X gate is applied; otherwise the input is unchanged. Repeating until $q=1$ gives an exact heralded correction with two attempts on average and almost-sure termination. It has no finite worst-case measurement count. A Pauli frame avoids this repeat-until-success procedure when only logical action or classical output statistics are required.

Measurement-based quantum computation prepares an entangled resource quantum state, then applies single-qubit quantum measurements, potentially adapting later bases to earlier outcomes. Pauli frame tracking and classical outcome processing convert random physical branches into a specified logical computation. Graph states and one-bit teleportation provide a concrete construction.

An explicit measurement-based quantum computation pattern uses six vertices $u_0,u_1,u_2,v_0,v_1,v_2$. Prepare a graph state with every vertex in $|+⟩$ and apply a Controlled-Z gate for each edgeThe first two links on each wire permit graph-state preparation of a computational-basis input followed by the logical J gate. Use the following single-qubit measurements:

All entangling edges can be made at preparation time because Controlled-Z gates commute. A future edge that does not touch a currently measured vertex can equivalently be deferred, which allows the one-bit teleportation identities to be applied in their logical order.

The two initial $X$ measurements implement $H|+⟩=|0⟩$ with Pauli frames $X^{s_0},X^{t_0}$ on the logical inputs. The first adaptive J gate then has output frame $X^{s_1}{Z}^{s_0}$ on $u_2$. Propagating through $E_{u_2{v}_1}$ gives framesup to branchwise global phase. The second adaptive J gate converts the latter intoA $Z$ correction does not alter a computational-basis measurement, while an $X$ correction flips its bit. Consequently the deterministic classical postprocessing isThis reproduces the output-bit distribution of the original quantum circuit, including its known byproduct corrections.

There is an additional simplification for these particular zero inputs. Since $J(\alpha )|0⟩=|+⟩$, $E(|+⟩|0⟩)=|+⟩|0⟩$, and $J(\beta )|0⟩=|+⟩$, the exact final state is $|+⟩|+⟩$, independently of the angles. The requested bit is therefore fair. A single isolated graph-state vertex measured in $Z$ already simulates that bit distribution; the six-vertex pattern also explicitly realizes the circuit and its corrections.

The J gate is $J(\alpha )=HP(\alpha )$, with $P(\alpha )=\mathrm{diag}(1,e^{i\alpha })$. Prepare a fresh qubit in $|+⟩$ and apply the Controlled-Z gate $E$ between it and the input $|\psi ⟩=u|0⟩+v|1⟩$. The resulting state isMeasure the input in the equatorial qubit measurement basis $|{\alpha }_s⟩=(|0⟩+(-1{)}^s{e}^{-i\alpha }|1⟩)/\sqrt{2}$. The unnormalized output isEach outcome has probability $1/2$. Thus one-bit teleportation realizeson the new qubit. Apply the known Pauli X gate correction $X^s$ for the literal $J(\alpha )$ output, or keep the correction in a Pauli frame and adapt later measurements. The old qubit is measured, so this is not cloning the input.

Direct multiplication of the given matrices gives $J(\alpha )X^s=e^{+is\alpha }{Z}^{s}J((-1{)}^s\alpha )$. The displayed negative exponent in the supplied relation has the wrong sign for exact matrix equality. The discrepancy is only a global phase in a fixed measurement branch, so it does not change this measurement implementation or its outcome probabilities. The positive-sign identity is used when tracking exact matrices.

The path graph state is $|{\psi }_3⟩=E_{23}{E}_{12}|+{⟩}_1|+{⟩}_2|+{⟩}_3$. The factor $Z_3^r$ commutes with both the Controlled-Z gates and the quantum measurement of vertex $1$. Treat $Z_1^r|+{⟩}_1$ as the input of one-bit teleportation. The equatorial qubit measurement result $s$ therefore gives the normalized stateHere the subscript on $Z_2^r$ expresses the teleported input factor; it is not an extra operation on the already measured vertex. Use $J(\alpha )Z^r=X^{r}J(\alpha )$ to obtainThe exclusive or exponent is equivalent to the printed sum $r+s$ because $X^2=I$. The conditional probability of $s$ is one half for either $r$, by the one-bit teleportation branch norm. This calculation concerns an equatorial measurement of a graph-state leaf; no third remaining qubit or extra $|-⟩$ factor is present.

The basic one-bit teleportation primitive uses the input as qubit 1 and a fresh $|+⟩$ quantum ancilla as qubit 2. Apply the Controlled-Z gate $E_{12}$ and measure qubit 1 in the equatorial qubit measurement basis with angle $\alpha$. If $s\in \{0,1\}$ is the result, the normalized post-measurement state on qubit 2 isBoth results have probability one half. For example, writing $|\psi ⟩=a|0⟩+b|1⟩$, the unnormalized branch is $(a|+⟩+(-1{)}^s{e}^{i\alpha }b|-⟩)/\sqrt{2}=X^{s}J(\alpha )|\psi ⟩/\sqrt{2}$. Thus $J(\alpha )$ is applied deterministically as a logical gate with known Pauli frame $X^s$; a permitted conditional Pauli X gate would remove this byproduct physically.

The literal resource restriction does not need an unstated direct Pauli X gate. A Pauli Z gate can be enacted by applying $E$ to the data and a fixed $|1⟩$ quantum ancilla. Two consecutive angle-zero one-bit teleportations, with results $p,q$, map an arbitrary current state $|\chi ⟩$ to $X^{q}H{X}^{p}H|\chi ⟩=X^q{Z}^p|\chi ⟩$. Apply the available $Z^p$ to remove the latter factor, up to an irrelevant global phase. This implements a heralded $X^q$: if $q=0$, the input is unchanged; if $q=1$, the required Pauli X gate has been applied.

If the original result is $s=1$ and a physically corrected output is required, repeat this heralded Pauli X correction using controlled-Z and measurements until $q=1$. Each attempt succeeds with probability one half independent of the input, so it terminates with probability one, using two attempts on average. Only $E$, fixed ancillary quantum states and single-qubit quantum measurements are used. This exact physical correction has no finite worst-case measurement bound; the standard finite deterministic realization is the logical Pauli frame version, which suffices for the later output-simulation parts.