For N > 4r, every 1D quantum cellular automaton of causality radius r is exactly a routed unitary circuit of nearest-neighbour interactions, and translation-invariant automata get translation-invariant circuits.
Fermionic cellular automata in one dimension
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abstract
We consider quantum cellular automata for one-dimensional chains of Fermionic modes and study their implementability as finite depth quantum circuits. Fermionic automata have been classified in terms of an index modulo circuits and the addition of ancillary systems. We strengthen this result removing the ancilla degrees of freedom in defining the equivalence classes. A complete characterization of nearest-neighbours automata is given. A class of Fermionic automata is found which cannot be expressed in terms of single mode and controlled-phase gates composed with shifts, as is the case for qubit cellular automata.
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Causal Decompositions of 1D Quantum Cellular Automata
For N > 4r, every 1D quantum cellular automaton of causality radius r is exactly a routed unitary circuit of nearest-neighbour interactions, and translation-invariant automata get translation-invariant circuits.