Information transport in 1D quantum cellular automata

Robert Raussendorf, Yiyang Gong

Gerhard Grössing, Anton Zeilinger

Shuichi Inokuchi, Kazumasa Honda, Hyen Yeal Lee +3 more

Bijections between sets may be seen as discrete (or crisp) unitary transformations used in quantum computations. So discrete quantum cellular automata are cellular automata with reversible transition functions. This note studies on 1d reversible cellular automata with triplet local rules.

Dirk-M. Schlingemann, Holger Vogts, Reinhard F. Werner

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.

Dirk-Michael Schlingemann

We report here on the structure of reversible quantum cellular automata with the additional restriction that these are also Clifford operations. This means that tensor products of Weyl operators (projective representation of a finite abelian symplectic group) are mapped to multiples of tensor products of Weyl 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 case we also find that all 1D Clifford quantum cellular automata are generated by a few elementary operations.

Pablo J. Arrighi