I think compared to the literature you’re using an overly restrictive and nonstandard definition of quantum cellular automata.
That makes sense! I’m searching for the simplest cellular-automaton-like thing that’s still interesting to study. I may have gone too far in the “simple” direction; but I’d like to understand why this highly-restricted subset of QCAs is too simple.
Specifically, it only makes sense to me to write Uas a product of operators like you have if all of the terms are on spatially disjoint regions.
Hmm! That’s not obvious to me; if there’s some general insight like “no operator containing two ~‘partially overlapping’ terms like ⋯⊗|x⟩(⟨x|+⟨y|)⊗|y⟩(⟨y|+⟨z|)⊗⋯ can be unitary,” I’d happily pay for that!
That makes sense! I’m searching for the simplest cellular-automaton-like thing that’s still interesting to study. I may have gone too far in the “simple” direction; but I’d like to understand why this highly-restricted subset of QCAs is too simple.
Hmm! That’s not obvious to me; if there’s some general insight like “no operator containing two ~‘partially overlapping’ terms like ⋯⊗|x⟩(⟨x|+⟨y|)⊗|y⟩(⟨y|+⟨z|)⊗⋯ can be unitary,” I’d happily pay for that!