 Solution

Write the final topological quantum order constants as $a>0$ and $\epsilon <1$, to avoid confusing the allowed support diameter with the small time coefficient. For any initial operator $A_X$ of operator norm at most one,and similarly for the partner state. The minus sign follows from $|{\psi }_0⟩=e^{itH}|{\psi }_1⟩$ and the Heisenberg picture convention $A_X(t)=e^{itH}{A}_X{e}^{-itH}$. The Lieb-Robinson bound and its localization corollary apply to either time direction, using $|t|$.

Choose $\mathrm{diam}(X)\le aL/4$, $t=\tau L$ with $v\tau \le a/8$, and localization buffer $l=aL/8$. The enlarged support $X^{\prime }$ obeysThe Lieb-Robinson localization by Haar twirling corollary supplies an operator $B_{X^{\prime }}$ approximating $A_X(-t)$ withDo not assume that the approximate operator has norm at most one: it only has $\Vert B_{X^{\prime }}\Vert \le 1+{\delta }_L$. Apply final-state local indistinguishability to $B_{X^{\prime }}/(1+{\delta }_L)$. The two approximation errors then giveIf the lattice has polynomially many sites in its diameter, $N(L)=O(L^p)$, the error tends to zero uniformly in these supports. For sufficiently large $L$, take ${\delta }_L\le (1-\epsilon )/[2(\epsilon +2)]$. The initial pair is then indistinguishable within ${\epsilon }^{\prime }=(1+\epsilon )/2<1$ on supports of diameter at most $aL/4$. Together with the exact orthogonality from condition (i), this proves the initial state is topologically ordered for sufficiently small linear-time coefficient. The constants for initial and final order need not be identical.

The backward preservation of local indistinguishability has an explicit finite-system qualification: it holds whenever the displayed $L|X|e^{-\mu aL/16}$ error is small enough. The conventional bounded-density, fixed-dimensional lattice interpretation supplies that condition. The question leaves growth control implicit; for an arbitrary collection of qudits one cannot discard the $|X|$ prefactor merely because $L$ is large. This identifies exactly the assumption needed by the supplied localization proof.