Aperiodicity in Quantum Wang Tilings

TL;DR

This paper generalizes Wang tilings to quantum and probabilistic settings, revealing how quantum interference can suppress periodicity and opening new research avenues in quantum automata and physics.

Contribution

It introduces a tensor-based quantum framework for Wang tiles, extending classical notions of tilings and aperiodicity, and demonstrates quantum effects on periodic patterns.

Findings

Quantum interference can suppress classical periodicity.

Decidability of the domino problem extends to one dimension.

Quantum aperiodic tile sets exist despite classical periodicity.

Abstract

By reformulating Wang tiles with tensors, we propose a natural generalization to the probabilistic and quantum setting. In this new framework, we introduce notions of tilings and periodicity directly extending their classical counterparts. In the one dimensional case, we recover the decidability of the generalized domino problem by linking it to the trace characterization of nilpotent matrices. In the two-dimensional case, we provide extension of weak and strong aperiodicity respectively and show the equivalence of those generalized notions, extending the well known equivalence in the classical case. We also exhibit a quantum tile set being aperiodic while its underlying classical tile set is not, proving that quantum interference can suppress periodic patterns and paving the way to the investigation of a new kind of aperiodicity. Finally, we highlight the many new research directions opened by this generalization of Wang tiles, related to (quantum) cellular automata, condensed matter physics, symbolic dynamics and more.