What is the Quantum Cheshire Cat experiments' import to Quantum Computing?
The quantum Cheshire Cat experiments appear to require postselection even to exhibit. Of course, postselection is itself a computational resource (and an extremely powerful one!) for "bounded" error quantum computation, and one whose power is characterised. To wit: the class PostBQP of decision problems decideable with bounded error using poly-size quantum circuits, conditioning on some single bit having the value 1 (provided that this occurs with non-zero probability) is precisely the class PP. (Recall also that PP contains NP as a subset.)

Perhaps one might ask whether quantum Cheshire cats might somehow represent a useful idiom in the design of  PP algorithms for difficult problems. Alternatively, perhaps quantum Cheshire cats correspond to some intriguing feature of quantum mechanics that can be described without resorting to (anything equivalent to) postselection, and which has implications for quantum computation. However, it isn't obvious at the moment that either of these are the case.