Causal Decompositions of 1D Quantum Cellular Automata

TL;DR

This paper proves that one-dimensional quantum cellular automata with a certain size can be decomposed into simpler, nearest-neighbor circuits, clarifying their causal structure and enabling better understanding of quantum causality.

Contribution

It establishes the first constructive causal decomposition for 1D QCAs with arbitrary causality radius, including translation-invariant cases, using novel algebraic techniques.

Findings

All 1D QCAs with N ≥ 4r + 1 admit causal decompositions.

Decompositions are translation-invariant for translation-invariant QCAs.

Provides a new algebraic framework for analyzing quantum causal structures.

Abstract

Understanding quantum theory's causal structure stands out as a major matter, since it radically departs from classical notions of causality. We present advances in the research program of causal decompositions, which investigates the existence of an equivalence between the causal and the compositional structures of unitary channels. Our results concern one-dimensional Quantum Cellular Automata (1D QCAs), i.e. unitary channels over a line of $N$ quantum systems (with or without periodic boundary conditions) that feature a causality radius $r$: a given input cannot causally influence outputs at a distance more than $r$. We prove that, for $N \geq 4r + 1$, 1D QCAs all admit causal decompositions: a unitary channel is a 1D QCA if and only if it can be decomposed into a unitary routed circuit of nearest-neighbour interactions, in which its causal structure is compositionally obvious. This provides the first constructive form of 1D QCAs with causality radius one or more, fully elucidating their structure. In addition, we show that this decomposition can be taken to be translation-invariant for the case of translation-invariant QCAs. Our proof of these results makes use of innovative algebraic techniques, leveraging a new framework for capturing partitions into non-factor sub-C* algebras.