Coarse-grained quantum cellular automata

TL;DR

This paper introduces a coarse-graining procedure for quantum cellular automata, showing convergence to a Dirac Hamiltonian and potential implementation on current quantum platforms, aiding understanding of systems with limited resolution.

Contribution

It provides a novel coarse-graining method for color-blind QCA following Goldilocks rules, connecting microscopic models to emergent effective dynamics.

Findings

Converges to a Dirac free Hamiltonian in the spacetime limit.

Introduces a discrete analog of Stokes law.

Potential for implementation on current quantum platforms.

Abstract

One can think of some physical evolutions as being the emergent-effective result of a microscopic discrete model. Inspired by classical coarse-graining procedures, we provide a simple procedure to coarse-grain color-blind quantum cellular automata that follow Goldilocks rules. The procedure consists in (i) space-time grouping the quantum cellular automaton (QCA) in cells of size $N$; (ii) projecting the states of a cell onto its borders, connecting them with the fine dynamics; (iii) describing the overall dynamics by the border states, that we call signals; and (iv) constructing the coarse-grained dynamics for different sizes $N$ of the cells. A byproduct of this simple toy-model is a general discrete analog of the Stokes law. Moreover we prove that in the spacetime limit, the automaton converges to a Dirac free Hamiltonian. The QCA we introduce here can be implemented by present-day quantum platforms, such as Rydberg arrays, trapped ions, and superconducting qbits. We hope our study can pave the way to a richer understanding of those systems with limited resolution.