Quantum causal graph dynamics

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. We discuss some of the implications for quantum gravity.

Figure 1

$K_u$ is a local unitary gate: it acts locally upon vertex $u$ and its neighbors, and may create superpositions of graphs. Consider global, unitary causal operator, which takes quantum superpositions of entire graphs into quantum superpositions of entire graphs—but in such a way that information propagates from one node to another only if they are neighbors in the graph. In this paper we show that unitary causal operators always decompose as a finite-depth circuit of such local unitary gates.