2008/04/28 by D. Schlingemann, Dirk-M. Schlingemann, Holger Vogts +1 · 48 citations
Computer Science · Mathematics · Physics and Astronomy · #Algorithm #Automata theory #Automaton #Cellular Automata and Applications #Cellular automaton #Computer science #Invariant (physics) #Mathematical Analysis and Transform Methods #Mathematical physics #Mathematics #Mobile automaton #Pure mathematics #Quantum cellular automaton #Quantum chaos and dynamical systems #Reversible cellular automaton #Stochastic cellular automaton #Tensor product #Theoretical computer science #quant-ph
paper · pdf · doi:10.1063/1.3005565
published in Journal of Mathematical Physics 49(11) (American Institute of Physics) · 28 pages, 2 figures, LaTeX
arxiv created 2008/04/28 · openalex publication_date 2008/11/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study reversible quantum cellular automata with the restriction that these are also Clifford operations. This means that tensor products of Pauli operators (or discrete Weyl operators) are mapped to tensor products of Pauli operators. Therefore Clifford quantum cellular automata are induced by symplectic cellular automata in phase space. We characterize these symplectic cellular automata and find that all possible local rules must be, up to some global shift, reflection invariant with respect to the origin. In the one-dimensional (1D) case we also find that every uniquely determined and translationally invariant stabilizer state can be prepared from a product state by a single Clifford cellular automaton time step, thereby characterizing this class of stabilizer states, and we show that all 1D Clifford quantum cellular automata are generated by a few elementary operations. We also show that the correspondence between translationally invariant stabilizer states and translationally invariant Clifford operations holds for periodic boundary conditions.