There are a couple of different models for using $n$ qubits to walk along a graph $G=(V,E)$.
- For example, we could assign each of the $n$ qubits to each of the $v\in V$ vertices, and have each edge $e\in E$ be a partial SWAP therebetween. Starting with a boson at one vertex and coloring the edges lets us walk the boson around the graph as $e^{-iHt}$. With only one boson, the traversal only explores the single-excitation subspace of the full $2^n$-dimensional Hilbert space. This is akin to a canonical (second) quantization of the walk along the graph, because each of the $n$ qubits records the presence or absence of the boson in that particular vertex (mode).
- Alternatively we can assign each $n$-bit string to a label of a vertex $v\in V$, and rely on oracles to tell us the adjacent vertices for the edge-coloring. Starting with a single boson we can explore a much larger graph, up to the full $2^n$-dimensional Hilbert space. I'll refer to this model as the first quantization of a continuous-time classical random walk, because here we explicitly note only the occupied vertex.
In both models we can ask about perfect state transfer (PST). For example, in the second-quantized model we can ask for a time $T$ such that $e^{-iHT}|10\cdots 00\rangle=|00\cdots 01\rangle$, while with first quantization we can ask for a time $T$ such that $e^{-iHT}|\text{source}\rangle=|\text{sink}\rangle.$
Perfect state transfer has all and everything to do with the eigenvalues of the underlying adjacency/Laplacian matrix. It is certainly useful in context of canonical quantization when we want to do long-range SWAPS of the contents of two registers. But is it ever useful to know that PST is established for an oracular, first-quantized graph problem? For example would it be useful to modify the conditions on the famous welded-trees problem so that the graph necessarily admits, or nearly admits, a perfect state transfer between the entrance vertex and the exit?
With welded trees, primacy is placed in the position of the particle in the graph (the mouse), while in canonical quantization, primacy is placed in the graph vertices themselves (the whole maze).
