One-Dimensional Quantum Cellular Automata over Finite, Unbounded Configurations

  • Authors:

  • Pablo Arrighi;Vincent Nesme;Reinhard Werner

  • Affiliations:

  • Université de Grenoble, LIG, Grenoble Cedex, France 38031;Technische Universität Braunschweig, IMAPH, Braunschweig, Germany 38106;Technische Universität Braunschweig, IMAPH, Braunschweig, Germany 38106

  • Venue:
  • Language and Automata Theory and Applications
  • Year:
  • 2008

Quantified Score

Hi-index 0.00

Visualization

Abstract

One-dimensional quantum cellular automata (QCA) consist in a line of identical, finite dimensional quantum systems. These evolve in discrete time steps according to a causal, shift-invariant unitary evolution. By causal we mean that no instantaneous long-range communication can occur. In order to define these over a Hilbert space we must restrict to a base of finite, yet unbounded configurations. We show that QCA always admit a two-layered block representation, and hence the inverse QCA is again a QCA. This is a striking result since the property does not hold for classical one-dimensional cellular automata as defined over such finite configurations. As an example we discuss a bijective cellular automata which becomes non-causal as a QCA, in a rare case of reversible computation which does not admit a straightforward quantization. We argue that a whole class of bijective cellular automata should no longer be considered to be reversible in a physical sense. Note that the same two-layered block representation result applies also over infinite configurations, as was previously shown for one-dimensional systems in the more elaborate formalism of operators algebras [13]. Here the proof is simpler and self-contained, moreover we discuss a counterexample QCA in higher dimensions.