I have a network problem that I believe would be represented as an adjacency-list on a quantum circuit. Typically, the list is up for traversing from various perspectives (e.g., shortest-path between two points).
Adjacency-list example:
x-> a, b, c
y-> a, d, e
x-> b, c, d
Any insight will be much appreciated!
Say you have $m$ vertices and the longest list you can build is of size $n$.
The simplest way I could think of is an encoding in basis state like : $$ | \text{origin} \rangle | v_0 (\text{origin})\rangle |v_1 (\text{origin})\rangle... |v_{n-1}(\text{origin})\rangle $$
We basically would encode the origin vertex and its corresponding n-sized adjacency-list whose elements are noted by $ v_0...v_{n-1} $. Potentially, the list may be of size less than $n$ but you can encode an empty element as a bitstring of your choice ($000$ for instance). Each vertex would be represented by a bit string of approximately $ \log(m+1) $ (qu)bits if you take into account the empty element.
In your example, the elements would be represented as $$(\text{empty},000),(a,001), (b,010), (c,011), (x,111), \cdots$$
For $x \to a,b,c$ you would have : $$ | x \rangle | a\rangle |b\rangle|c\rangle = | 111 \rangle | 001\rangle |010\rangle|011\rangle = |111 001 010 011 \rangle$$
That means also you can have all pairs of $(\text{origin, list})$ in superposition, where at the end you retrieve one by measurement according to the corresponding amplitudes.
