In quantum computing, determining the code distance of a stabilizer code is similar to the shortest path problem on a hypergraph. Each node in the graph would be some sort of parity check performed by the code, and each hyperedge would be a set of parity checks a particular physical error flips. If you construct the graph correctly, paths between certain nodes will correspond to logical errors and the code distance will correspond to a shortest path because the shortest path corresponds to the minimum number of errors. So that's the motivation for being interested in the shortest path problem on a hypergraph.
I did some preliminary searching for existing algorithms that solve this problem. Unfortunately, it seems that existing research is actually referring to a slightly different problem than what I had in mind. In particular, existing work seems to be happy to pass through edges that touch unrelated nodes an odd number of times. In the context of a code distance calculation this would be a bad idea, because it would cause parity check violations elsewhere in the graph. The "minimum length path" wouldn't successfully get back into the desired codespace.
The variant of the shortest hyperpath problem that I care about is as follows. Let $n$ and $m$ be the start and end nodes. Let $E$ be a set of hyperedges. $E$ is a hyperpath between $n$ and $m$ if $n$ appears in an odd number of the hyperedges of $E$, $m$ also appears in an odd number of the hyperedges of $E$, and all other nodes appear an even number of times in the hyperedges of $E$. I want $n$ and $m$ to be the only nodes whose corresponding parity check triggers when the corresponding errors from $E$ are applied. The goal is to find the hyperpath with the minimum number of hyperedges.
Has there been previous work on this problem? Note that I'm mostly interested in the case where the hyperedge degree is bounded (e.g. the error graph of the surface code typically has no hyperedges with degree larger than 4, even with circuit level noise and two qubit gate errors).