Coined Quantum Walks as Quantum Markov Chains

We analyze the equivalence between discrete-time coined quantum walks and Szegedy’s quantum walks. We characterize a class of flip-flop coined models with generalized Grover coin on a graph $\mathrm{\Gamma }$ that can be directly converted into Szegedy’s model on the subdivision graph of $\mathrm{\Gamma }$ and we describe a method to convert one model into the other. This method improves previous results in literature that need to use the staggered model and the concept of line graph, which are avoided here.