= Solution
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 $|+\rangle$ and apply a <Controlled-Z gate> for each edge
$$
(u_0,u_1),\ (u_1,u_2),\ (v_0,v_1),\ (v_1,v_2),\ (u_2,v_1).
$$
The 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:
* Measure $u_0$ and $v_0$ in the $X$ basis, obtaining $s_0,t_0$.
* Measure $u_1$ in the equatorial basis with angle $(-1)^{s_0}\alpha$, obtaining $s_1$.
* Measure $v_1$ in the equatorial basis with angle $(-1)^{t_0}\beta$, obtaining $t_1$.
* Measure $v_2$ in the $Z$ basis, obtaining $m$, and return $b_2=m\oplus s_1\oplus t_1$. The unmeasured $u_2$ can be discarded.
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 $H|+\rangle=|0\rangle$ 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 $E_{u_2v_1}$ gives frames
$$
X^{s_1}Z^{s_0\oplus t_0}\ \text{on }u_2,
\qquad X^{t_0}Z^{s_1}\ \text{on }v_1,
$$
up to branchwise global phase. The second adaptive <J gate> converts the latter into
$$
X^{t_1\oplus s_1}Z^{t_0}\ \text{on }v_2.
$$
A $Z$ correction does not alter a computational-basis measurement, while an $X$ correction flips its bit. Consequently the deterministic classical postprocessing is
$$
\boxed{b_2=m\oplus s_1\oplus t_1}.
$$
This reproduces the output-bit distribution of the original <quantum circuit>, including its known byproduct corrections.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/iii/paper-61-measurement-pattern.png]
{title=Six-vertex graph state, adaptive equatorial measurements and classical parity correction for the two-wire circuit}
{height=480}
There is an additional simplification for these particular zero inputs. Since $J(\alpha)|0\rangle=|+\rangle$, $E(|+\rangle|0\rangle)=|+\rangle|0\rangle$, and $J(\beta)|0\rangle=|+\rangle$, the exact final state is $|+\rangle|+\rangle$, 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.
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