Quantum Walks with Indefinite Causal Order

In all existing quantum walk models, the assumption about a pre-existing fixed background causal structure is always made and has been taken for granted. Nevertheless, in this work we will get rid of this tacit assumption especially by introducing indefinite causal order ways of coin tossing and investigate this modern scenario. We find that an ideal-shape and fast-spreading uniform distribution can be prepared with our new model. First we will show how an always-symmetrical instantaneous distribution appears in an indefinite causal order quantum walk, which then paves the way for deriving conditions that enables one to interpret an evolved state as a superposition of its definite causal order counterparts which is however in general prohibited. This property of temporal superposition of states can be generalized from two-process scenarios to cases where arbitrary many $\mathcal{N}$ processes are involved. Finally, we demonstrate how a genuine uniform distribution emerges from an indefinite causal order quantum walk. Remarkably, besides the ideal shape of the distribution, our protocol has another powerful advantage, that is the speed of spatial spreading reaches exactly the theoretical limit in contrast to conventional cases where one may encounter the well-known issue of a $1/\sqrt{2}$ degeneration.