I'm trying to understand how the Szegedy quantum walk operator is useful. I understand that given a reducible ergodic Markov chain with transition matrix $P\in\mathbb{R}^{2^n\times 2^n}$, implemented as an oracle $O$, we can implement a unitary $U\in\mathrm{U}(2n)$ using $O$ and standard gates (swaps and reflections) such that $U$ is a block-encoding of a matrix similar to $P$, and moreover if $\lambda_j$ are $P$'s eigenvalues then $U$ has eigenvalues $e^{\pm i\arccos\lambda_j}$. This realizes an angular spectral gap where the complex eigenvalue nearest to 1 is $e^{\pm i\arccos\lambda_2}$ at angular distance $O(\sqrt{1-\lambda_2})$ (using the convention that $1=\lambda_1>\lambda_2\ge\cdots$) (and assuming for convenience that $\lambda_2\ge-\lambda_{2^n}$ (WLOG replace $P$ with $\frac{P+I}{2}$)).
The application that I have seen of this construction is to identifying if a vertex in $K_{2^n}$ is marked with query cost $\sqrt{2^n}$ to $O$, where the idea is to use qubitization. However, this requires an extremely specialized setup where the two stochastic matrices being compared have spectra $\{0,\dots,0,1\}$ and $\{0,\dots,0,1-\frac{1}{2^n},1\}$, and the analysis strikes me as a little delicate with this regard.
It would seem that the advantage to having an angular spectral gap of $c\sqrt{\gamma}$ against spectral gap of $\gamma$ is that in $\frac{\pi/2}{c\sqrt{\gamma}}$ calls, the second eigenvalue gets "pushed away" on the circle from 1 all the way over to $i$ -- but we don't have any control on, say, the third eigenvalue, which in this sequence of calls may wrap back around to being near to 1. So I guess my main question is how the quadratic speedup is actually realizable in sampling from Markov chain stationary distributions in general in this setup. (I think I have seen some discussion of related ideas online but have gotten rather spun around.)
Thanks in advance, and let me know if I can clarify any of the setup here.
