As an extension of a famous description of non-local Hamiltonian simulation in section 4.7.3 of Nielsen and Chuang, the welded-trees paper of Childs, et al. provides the following circuit for use in the quantum walk along their graph:
Here, $a$ and $b$ are two different $2n$-bit labels for the nodes of the graph, while $W$ appears somehow related to a SWAP gate used in the traversal of the graph.
In particular, the eigenvalues of SWAP are $\pm 1$ because SWAP$\ne$SWAP$^2=\mathbb 1$, while the unique eigenvector of the SWAP gate having eigenvalue $-1$ is $\frac{1}{\sqrt 2}(|01\rangle-|10\rangle)$.
Childs et al. define $W$ as:
$$\begin{eqnarray} W|00\rangle=|00\rangle\\ W\frac{1}{\sqrt 2}(|01\rangle+|10\rangle)=|01\rangle\\ W\frac{1}{\sqrt 2}(|01\rangle-|10\rangle)=|10\rangle\\ W|11\rangle=|11\rangle \end{eqnarray}$$
But what, then, are the entries of this gate $W$?
Are they something like: $$\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & e^{i\pi q/2}\cos\frac{\pi q}{2} & -ie^{i\pi q/2}\sin\frac{\pi q}{2} & \: 0 \ \\ 0 & -ie^{i\pi q/2}\sin\frac{\pi q}{2} & e^{i\pi q/2}\cos \frac{\pi q}{2} & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) $$
for some $q$?
I'm really just trying to understand that paper in general, and this circuit in particular, in more detail. Childs et al. trotterizes this circuit over all the different edge-colorings of the graph to do the Hamiltonian simulation thereof; after about a linear amount of time there's a decent odds that the particle, which was initially at node $|\psi\rangle=$ ENTRY, has found the EXIT.
