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Pauli Center for Theoretical Studies

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Quantum Science and Technology

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Sandia National Laboratories

Institute for Quantum Computing

id Quantique

Unitarity plus causality implies localizability

Vincent Nesme, Leibniz

joint work with Pablo Arrighi and Reinhard Werner

We consider a graph with a single quantum system at each node. The entire compound system evolves in discrete time steps by iterating a global evolution U. We require that this global evolution U be unitary, in accordance with quantum theory, and that this global evolution U be causal, in accordance with special relativity. By causal we mean that information can only ever be transmitted at a bounded speed, the speed bound being quite naturally that of one edge of the underlying graph per iteration of U. We show that under these conditions the operator U can be implemented locally; i.e. it can be put into the form of a quantum circuit made up with more elementary operators -each acting solely upon neighbouring nodes. We take quantum cellular automata as an example application of this representation theorem: this analysis bridges the gap between the axiomatic and the constructive approaches to defining QCA. The longer version of this extended abstract is available as arXiv:0711.3975.