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We set the criteria for a quantum walk to exhibit nontrivial dynamics whenplaced in an indefinite causal order and study two-period quantum walks whenthe evolution operators are arranged in a causally ordered sequence and in anindefinite causal order using quantum switch. When either forward or backwardcausal sequence is implemented, one observes a causal asymmetry in thedynamics, in the sense that the reduced dynamics of the coin state is morenon-Markovian for one particular temporal order of operations than that of theother. When the dynamics is defined using evolution operators in asuperposition of causal orders, the reduced dynamics of the coin space exhibithigher non-Markovian behavior than either of the definite causal orders. Thiseffect can be interpreted as a Parrondo-like effect in non-Markovianity of thereduced state dynamics of the coin. We further generalize the qualitativedescription of our results pertaining to the dynamics when the walk has ahigher number of periods.

Superposition of causal order in quantum walks: non-Markovianity and causal asymmetry