= Solution
For short times, survival under <unitary time evolution> is $1-(\Delta H)^2t^2/\hbar^2+O(t^3)$. Dividing a fixed time into $N$ intervals and projecting back after each gives survival approaching one as $N\to\infty$. This suppression of departure by repeated <projective measurements> is the <quantum Zeno effect>.
For <measurement-guided Zeno transfer>, rotate the measured basis gradually to transport a known state. Let
$$
|b_j\rangle=\cos\theta_j|0\rangle+\sin\theta_j|1\rangle,\quad
|b_j^\perp\rangle=-\sin\theta_j|0\rangle+\cos\theta_j|1\rangle,\quad
\theta_j=\frac{j\pi}{2N}.
$$
Measure in these orthonormal bases successively for $j=1,\ldots,N$. Starting in $|0\rangle=|b_0\rangle$, the probability to follow the $b_j$ outcome at every step is
$$
\boxed{P_{\rm follow}=\cos^{2N}\left(\frac{\pi}{2N}\right)\longrightarrow1.}
$$
Its final state is $|b_N\rangle=|1\rangle$. If every outcome is retained without postselection, the final probability for $|1\rangle$ is $[1+\cos^N(\pi/N)]/2$, also approaching one; histories with two switches can return to that branch. These two finite-$N$ probabilities should not be confused.
For a general unknown input, the same slowly turning measurements first resolve its amplitudes into the two branches, then drag each branch separately. In the large-$N$ limit, after discarding the records,
$$
\boxed{\rho_{\rm out}\longrightarrow|\alpha|^2|1\rangle\langle1|+|\beta|^2|0\rangle\langle0|.}
$$
The first basis approaches the computational one as $N$ grows. Inserting an initial computational measurement makes this branch decomposition exact at the outset, without changing the limiting result. Relative phase is lost, so this procedure transports a known basis state with high probability rather than implementing a coherent <Pauli X gate> on an arbitrary <qubit>.
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