Physics – Quantum Physics
Scientific paper
2007-11-26
QIP 2010
Physics
Quantum Physics
V1: 5 pages, revtex. V2: Generalizes V1, explanatory slides available at http://membres-lig.imag.fr/arrighi/ucausaltalk/QICS08
Scientific paper
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. KEYWORDS: Quantum cellular automata, Unitary causal operators, Quantum walks, Quantum computation, Axiomatic quantum field theory, Algebraic quantum field theory, Discrete space-time.
Arrighi Pablo
Nesme Vincent
Werner Reinhard
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