Let's start with a simple example where $H_i$ and $H_f$ commute because they are both diagonal:

$H_i=
\begin{pmatrix}1 & 0\\
0 & -1
\end{pmatrix}
$

$H_p=
\begin{pmatrix}-1 & 0\\
0 & -0.1
\end{pmatrix}
$

The eigenvector with lowest eigenvalue (i.e. the ground state) of $H_i$ is $|1\rangle $ so we start in this state.
The ground state of $H_f$ is $|0\rangle$ so this is what we're looking for.

Remember the minimum runtime for the AQC to give the correct answer to within an error $\epsilon$:

$\tau\ge \max_t\left(\frac{||H_i - H_f||^2}{\epsilon E_{\rm{gap}}(t)^3}\right)$.

This is given and explained in Eq. 2 of Tanburn *et al.* (2015).

- Let's say we want $\epsilon = 0.1$.
- Notice that $||H_i - H_f||^2 = 0.1 $ according Eq. 4 of the same paper.
- Notice that $\frac{||H_i - H_f||^2}{\epsilon}=1$ (I've chosen $\epsilon$ so that this would happen, but it doesn't matter).
- We now have $\tau \ge \max_t\left(\frac{1}{E_{\rm{gap}}(t)^3}\right)$

So what is the minimum gap between ground and first excited state (which gives the $\max_t$) ?

When $t=20\tau/29$, the Hamiltonian is:

$H=\frac{9}{29}H_i + \frac{20}{29}H_p$

$H=\frac{9}{29}\begin{pmatrix}1 & 0\\
0 & -1
\end{pmatrix} + \frac{20}{29}\begin{pmatrix}-1 & 0\\
0 & -0.1
\end{pmatrix}$

$
H=\begin{pmatrix}\frac{9}{29} & 0\\
0 & -\frac{9}{29}
\end{pmatrix}+\begin{pmatrix}-\frac{20}{29} & 0\\
0 & -\frac{2}{29}
\end{pmatrix}
$

$
H=\begin{pmatrix}\frac{-11}{29} & 0\\
0 & -\frac{11}{29}
\end{pmatrix}
$

So when $t=\frac{20}{29}\tau$, we have $E_{\rm{gap}}=0$ and the *lower bound* on $\tau$ is essentially $\infty$.

So the adiabatic theorem still applies, but when it states that the Hamiltonian needs to change "slowly enough", it turns out it needs to change "infinitely slowly", which means you will not likely ever get the answer using AQC.

1This sounds quite convincing and clear. May you explicitly explain why there cannot be an avoided crossing during the adiabatic evolution (which would allow the nature of the ground state to change but with no degeneracy)? – agaitaarino – 2018-04-27T14:50:52.693