Unitarity plus causality implies localizability

TL;DR

This paper proves that a quantum system evolving unitarily and causally on a graph can be implemented as a local quantum circuit, bridging axiomatic and constructive approaches in quantum cellular automata.

Contribution

It demonstrates that unitarity and causality together imply the local implementability of the global evolution operator in quantum systems.

Findings

Global evolution U can be realized as a local quantum circuit.

Causality constrains information transfer to bounded speed.

Application to quantum cellular automata illustrates the theoretical result.

Abstract

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.