My question concerns a proof in https://people.eecs.berkeley.edu/~vazirani/pubs/qao.pdf, "Limits on Quantum Adiabatic Optimization - Warning: Rough Manuscript!" by Wim van Dam and Umesh Vazirani. It seems that it was never completed, although it has been cited several times since. In particular, Lemma 2 states (summarized and paraphrased):
If a quantum adiabatic algorithm is carried out on a graph of size $n$ with diameter $w$, with a Hamiltonian whose entries are polynomial in $\log(n)$, the optimization will take time $2^{(w \log w)}$
This is a bit interpretation on my part, because the proof is very incomplete and inconsistent. Now, if this statement is true, we could* apply it to an expander graph with degree $d = \log^k(n)$, and get a graph of width $w = c\log(n)/\log(d)$ for some constant $c$.
That would imply a runtime of
$$d 2^{w\log w} = \log^k(n) 2^{(c \log n/k \log\log n) (\log\log n)} \le \log^k(n) 2^{(c \log n/k)}$$
Then by taking $k$ to be sufficiently large, say, $4/c$, this gives you a $poly(log(n))*2^{\log(n)/4}$, which appears to outperform Grover's algorithm, which is strong evidence that something here has gone wrong. :) As pointed out in the second answer to this question, there might currently be some loopholes to allow adiabatic computing to outperform gate models, but that seems extremely unlikely -- I assume I am not understanding the Lemma 2 from van Dam's manuscript, or the relevant expander graphs don't exist somehow (e.g. $c$ would depend on $n$ in some nontrivial way or something.)
Can anyone explain what's going on here, and perhaps provide a proof of that Lemma 2?
[*] This requires the existence of a such a graph. Moore graphs seem sufficient (a degree-$d$ tree branching out for $w$ levels). Ramanujan graphs might work, if they exist small enough; for example, for the case of Ramanujan graphs, the second-largest eigenvalue is bounded by $\lambda_2 \le 2\sqrt{d-1}$, and the edge expansion is lower-bounded by $\frac{1}{2}(d-\lambda_2) \ge \frac{d}{3}$. A graph with $n$ vertices and edge expansion $\phi$ has diameter bounded by $2\log(n)/\log(1+\phi)$.
