Many paradoxes of quantum mechanics come from the fact that a quantum system can possess different features at the same time, such as in wave-particle duality or quantum superposition. In recent delayed-choice experiments, a quantum mechanical system
can be observed to manifest one feature such as the wave or particle nature, depending on the final measurement setup, which is chosen after the system itself has already entered the measuring device; hence its behaviour is not predetermined. Here, we adapt this paradigmatic scheme to multi-dimensional quantum walks. In our experiment, the way in which a photon interferes with itself in a strongly non-trivial pattern depends on its polarisation, that is determined after the photon has already been detected. Multi-dimensional quantum walks are a very powerful tool for simulating the behaviour of complex quantum systems, due to their versatility. This is the first experiment realising a multi-dimensional quantum walk with a single-photon source and we present also the first experimental simulation of the Grover walk, a model that can be used to implement the Grover quantum search algorithm.
