Quantum Walks under Superposition of Causal Order

We set the criteria under which superposition of causal order can be incorporated into quantum walks. In particular, we show that only periodic quantum walks or those with at least one disorder exhibit Superposition of causal order under the action of ‘quantum switch’. We exemplify our results with a simple example of two-period discrete-time quantum walks. In particular, we observe that periodic quantum walks exhibit causal asymmetry pertaining to the dynamics of the reduced coin state: the dynamics are more non-Markovian for one temporal order than the other. We also note that the non-Markovianity of the reduced coin state due to indefiniteness in causal order tends to match the dynamics of a particular temporal order of the coin state. We substantiate our results with numerical simulations.