One-dimensional quantum cellular automata over finite, unbounded configurations

TL;DR

This paper proves that one-dimensional quantum cellular automata (QCA) over finite, unbounded configurations always have a two-layered block representation, ensuring their inverse is also a QCA, which is a novel result contrasting classical automata.

Contribution

It establishes a universal two-layered block representation for 1D QCA over finite configurations and shows the inverse is also a QCA, unlike classical automata.

Findings

QCA always admit a two-layered block representation

Inverse QCA is also a QCA for 1D systems

Counterexample of higher-dimensional QCA discussed

Abstract

One-dimensional quantum cellular automata (QCA) consist in a line of identical, finite dimensional quantum systems. These evolve in discrete time steps according to a local, shift-invariant unitary evolution. By local we mean that no instantaneous long-range communication can occur. In order to define these over a Hilbert space we must restrict to a base of finite, yet unbounded configurations. We show that QCA always admit a two-layered block representation, and hence the inverse QCA is again a QCA. This is a striking result since the property does not hold for classical one-dimensional cellular automata as defined over such finite configurations. As an example we discuss a bijective cellular automata which becomes non-local as a QCA, in a rare case of reversible computation which does not admit a straightforward quantization. We argue that a whole class of bijective cellular automata should no longer be considered to be reversible in a physical sense. Note that the same two-layered block representation result applies also over infinite configurations, as was previously shown for one-dimensional systems in the more elaborate formalism of operators algebras [9]. Here the proof is made simpler and self-contained, moreover we discuss a counterexample QCA in higher dimensions.