A decision procedure for unitary linear quantum cellular automata

Linear quantum cellular automata were introduced recently as one of the models of quantum computing. A basic postulate of quantum mechanics imposes a strong constraint on any quantum machine: it has to be unitary; that is, its time evolution operator has to be a unitary transformation. In this paper we give an efficient algorithm to decide if a linear quantum cellular automaton is unitary. The complexity of the algorithm is O(n((3r-1)/(r+1))) = O(n(3)) in the algebraic computational model if the automaton has a continuous neighborhood of size r, where n is the size of the input.