Given two digraphs $G$ and $H$, I want a method for creating a bijection between all non-simple cycles of for all $n \le |V(G)|$. That means, given $C_G(n)$ and $C_H(n)$ being the sets of all non-simple cycles of length $n$ in $G$ and $H$ respectively, there is a bijection between these two sets. Here, non-simple cycles refers to a closed walk on the graph where we are allowing repeated vertices and edges.

First, it is known that tr$(A^n)$ where $A$ is the adjacency matrix of a graph gives the number of non-simple cycles of length $n$. So, in polynomial time, it is possible to determine whether such a bijection can exist between graphs $G$ and $H$. The question is how to find this bijection without simply enumerating all the possible cycles and pairing them off one by one which is at least an exponential algorithm.

My first thought was to use a cycle basis. However, when considering non-simple cycles, there isn't a unique vector representation in $\mathbb{Q}^{|E(G)|}$ since edges can repeat.

Are there any other representations of the cycles in a graph besides a cycle basis that could reduce the number of cycles to be considered to a polynomial amount?