  Coherent evolution governs the behaviour of all quantum systems, but in
nature it is often subjected to influence of a classical environment. For
analysing quantum transport phenomena quantum walks emerge as suitable model
systems. In particular, quantum walks on percolation structures constitute an
attractive platform for studying open system dynamics of random media. Here, we
present an implementation of quantum walks differing from the previous
experiments by achieving dynamical control of the underlying graph structure.
We demonstrate the evolution of an optical time-multiplexed quantum walk over
six double steps, revealing the intricate interplay between the internal and
external degrees of freedom. The observation of clear non-Markovian signatures
in the coin space testifies the high coherence of the implementation and the
extraordinary degree of control of all system parameters. Our work is the
proof-of-principle experiment of a quantum walk on a dynamical percolation
graph, paving the way towards complex simulation of quantum transport in random
media.
