Let $G$ be an unweighted, undirected graph on $n$ vertices with vertex labels $\{1,2,\cdots,n\}$; let $A$ be its $n\times n$ adjacency matrix. We can prepare a state $|\psi\rangle$ representing $A$ with $n^2$ qubits; for example, with $n=10$ we could have the famous Petersen graph with an adjacency matrix $A$ as:
$$A=\begin{pmatrix} 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 \end{pmatrix},\tag 1$$
where $|\psi\rangle=|A\rangle.$
I wish to construct a continuous-time quantum walk $U=\exp(-\mathrm iHt)$ that acts unitarily on $|\psi\rangle$, moving along different permutations of the vertices. My walk should stay within the subspace spanned by all $n\times n$ adjacency matrices that are isomorphic to $A$. $H$ will then correspond to a larger graph, with $H$'s vertices corresponding to adjacency matrices and $H$'s edges corresponding to permutations between adjacency matrices.
To this end, consider the symmetric group $S_n$ of all permutations of $\{1,2,\cdots, n\}$. We can envision a swap-like gate, $\text{SWAP}_{(jk)}$, that when acting on $|A\rangle$ swaps row/column $j$ with row/column $k$; let's focus for now on the generating set $(12)(13)\cdots(1n)$ having $n-1$ such generators.
For example, $\text{SWAP}_{(13)}|A\rangle$ is the permuted adjacency matrix
$$A'= \begin{pmatrix} 0& 0& 1& 1& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 1& 1& 0& 0& 0& 0\\ 1& 0& 0& 0& 1& 0& 1& 0& 0& 0\\ 1& 1& 0& 0& 0& 0& 0& 0& 0& 1\\ 0& 1& 1& 0& 0& 0& 0& 0& 1& 0\\ 0& 1& 0& 0& 0& 0& 1& 1& 0& 0\\ 0& 0& 1& 0& 0& 1& 0& 0& 0& 1\\ 1& 0& 0& 0& 0& 1& 0& 0& 1& 0\\ 0& 0& 0& 0& 1& 0& 0& 1& 0& 1\\ 0& 0& 0& 1& 0& 0& 1& 0& 1& 0 \end{pmatrix}.\tag 2$$
Noting that in addition to $\text{SWAP}_{(jk)}$ being unitary, it is also an involution. Accordingly, we have:
$$\sqrt[m]{\text{SWAP}_{(jk)}}=\exp\left(\mathrm{i}\frac{\pi}{2m}(I-\text{SWAP}_{(jk)})\right).\tag 3$$
Such a circuit for the $m$-th root of a swap between rows and columns $j$ of $A$ with rows and columns $k$ of $A$ is easy to construct.
From here we see that, cycling through each generator $(12),(13),\cdots,(1n)$, and repeating $m$ times lets us unitarily evolve over the Hilbert space spanned by adjacency matrices - as if we are evolving a quantum along the Cayley graph of $S_n$ modded out by the automorphism group of $G$ and generated by $(12),(13),\cdots,(1n),$ according to Schrödinger's equation.
Can we learn anything novel about $G$ with such an approach? For example, if we were to evolve by cycling:
$$U=\left(\prod_{j\ne 1}^n\sqrt[m]{\text{SWAP}_{(1j)}}\right)^m\approx\exp\left(\mathrm i\frac{\pi}{2}\mathcal L\right),\tag 4$$
where:
$$\mathcal L=(n-1)I-\sum_{j\ne 1}^n\text{SWAP}_{(1j)},\tag 5$$
is the Laplacian of the underlying Cayley graph of $S_n/\rm{Aut}(G)$ can we say anything about where we would end up after, say, a polynomial number $U^{\rm{poly}}|\psi\rangle$ of applications of $U$?
How about if we prepare, not $|\psi\rangle=|A\rangle$, but rather $|\psi\rangle=\frac{1}{\sqrt 2}\big(|A\rangle-|B\rangle\big)$ for two separate adjacency matrices of $G$, and have $U$ act on $|\psi\rangle$ a polynomial number of times? Or choose a different generating set, such as $(12),(23),\cdots,(n-1\: n)$?
For an $n$-vertex graph, we use $O(n-1)$ generators to walk along the Cayley graph by using the above product formula - we could also use the more natural generating set $(jk)$ (see John Watrous's comment below), which would have $O(n^2)$ generators; I guess we could even use only two generators such as $(12),(23\cdots n)$ but the second generator $(23\cdots n)$ is not an involution so Equation $3$ doesn't apply.
Certainly this seems like a pretty natural way of constructing a Hamiltonian to explore graph isomorphism-related problems. Perhaps running a phase estimation circuit on various states tells us interesting properties thereof?
