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.
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Arrighi, P., Nesme, V., Werner, R. (2008). One-Dimensional Quantum Cellular Automata over Finite, Unbounded Configurations. In: Martín-Vide, C., Otto, F., Fernau, H. (eds) Language and Automata Theory and Applications. LATA 2008. Lecture Notes in Computer Science, vol 5196. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88282-4_8
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DOI: https://doi.org/10.1007/978-3-540-88282-4_8
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