There are many articles on quantum cellular automata. See for example “A review of Quantum Cellular Automata”, or “Quantum Cellular Automata, Tensor Networks, and Area Laws”. I think compared to the literature you’re using an overly restrictive and nonstandard definition of quantum cellular automata. Specifically, it only makes sense to me to write U as a product of operators like you have if all of the terms are on spatially disjoint regions.
Consider defining quantum cellular automata instead as local quantum circuits composed of identical two-site unitary operators everywhere:
If you define them like this, then basically any kind of energy and momentum conserving local quantum dynamics can be discretized into a quantum cellular automata, because any two-site time and space independent quantum Hamiltonian can be decomposed into steps with identical unitaries like this using the Suzuki-Trotter decomposition.
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!
There are many articles on quantum cellular automata. See for example “A review of Quantum Cellular Automata”, or “Quantum Cellular Automata, Tensor Networks, and Area Laws”.
I think compared to the literature you’re using an overly restrictive and nonstandard definition of quantum cellular automata. Specifically, it only makes sense to me to write U as a product of operators like you have if all of the terms are on spatially disjoint regions.
Consider defining quantum cellular automata instead as local quantum circuits composed of identical two-site unitary operators everywhere:
If you define them like this, then basically any kind of energy and momentum conserving local quantum dynamics can be discretized into a quantum cellular automata, because any two-site time and space independent quantum Hamiltonian can be decomposed into steps with identical unitaries like this using the Suzuki-Trotter decomposition.
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!