Quantum Causal Graph Dynamics

• Published in: arXiv.org
• Main Authors: Arrighi, Pablo, Martiel, Simon
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 02.06.2017
• Subjects: Cellular automata; Computer Science - Discrete Mathematics; Gates (circuits); Graphical representations; Graphs; Physics - General Relativity and Quantum Cosmology; Physics - Quantum Physics; Quantum gravity; Quantum mechanics; Superposition (mathematics)
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1607.06700

Consider a graph having quantum systems lying at each node. Suppose that the whole thing evolves in discrete time steps, according to a global, unitary causal operator. By causal we mean that information can only propagate at a bounded speed, with respect to the distance given by the graph. Suppose, moreover, that the graph itself is subject to the evolution, and may be driven to be in a quantum superposition of graphs---in accordance to the superposition principle. We show that these unitary causal operators must decompose as a finite-depth circuit of local unitary gates. This unifies a result on Quantum Cellular Automata with another on Reversible Causal Graph Dynamics. Along the way we formalize a notion of causality which is valid in the context of quantum superpositions of time-varying graphs, and has a number of good properties. Keywords: Quantum Lattice Gas Automata, Block-representation, Curtis-Hedlund-Lyndon, No-signalling, Localizability, Quantum Gravity, Quantum Graphity, Causal Dynamical Triangulations, Spin Networks, Dynamical networks, Graph Rewriting.