Gap closing invalidates a uniform adiabatic bound

The path is linear in $s$, hence twice continuously differentiable. Since $Z=\mathrm{diag}(1,-1)$, its eigenvalues are $s-1\mathrm{/}2$ and $-(s-1\mathrm{/}2)$; the gap is $g(s)=2\mathrm{\mid }s-1\mathrm{/}2\mathrm{\mid }$, which vanishes at $s=1\mathrm{/}2$. Therefore no positive uniform full gap exists.

At $s=0$, $H(0)=-Z\mathrm{/}2$ has unique ground vector $\mathrm{\mid }0⟩$; at $s=1$, $H(1)=Z\mathrm{/}2$ has unique ground vector $\mathrm{\mid }1⟩$. The Hamiltonians commute for all times and are diagonal, so the solution of $i{\mathrm{\partial }}_s{U}_T=TH(s)U_T$ maps $\mathrm{\mid }0⟩$ to a scalar phase times $\mathrm{\mid }0⟩$. Its final overlap with the final ground vector is zero, and its transition probability out of the final ground space is exactly $1$ for every $T>0$.

The path satisfies the smoothness assumption but violates the strictly positive uniform-gap hypothesis in Finite JRS adiabatic projector bound. Step 1.2 shows directly that the desired ground-state-following conclusion can fail completely when that hypothesis is removed.