Quantum Causal Graph Dynamics

TL;DR

This paper demonstrates that quantum causal graph dynamics can be represented as finite-depth circuits of local unitary gates, unifying quantum cellular automata and causal graph dynamics in a superposition framework.

Contribution

It introduces a formal framework for quantum causal graph dynamics, showing that such dynamics decompose into local unitary circuits and formalizing causality in superposed, evolving graphs.

Findings

Quantum causal operators decompose into finite-depth local circuits.

A formal notion of causality for superposed, time-varying graphs is established.

Unification of Quantum Cellular Automata and Reversible Causal Graph Dynamics.

Abstract

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.