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 procedures to coarse-grain color-blind quantum cellular automata, following Goldilock rules. The procedure consists in (i) space-time grouping the quantum cellular automaton (QCA) in cells of size $N$, (ii) projecting the cells states onto its borders, connecting them with the fine dynamics, which is reminiscent of the edge-bulk correspondence in quantum field theory; (iii) describe the overall dynamics by the border states, that we call signals; (iv) construct the coarse-grained dynamics for different sizes $N$ of the cells. A byproduct of this simple toy-model is a general discrete analogous of the Stokes law. Moreover, we proved that in the spacetime limit, the automaton converge to a Dirac free Hamiltonian. The QCA we introduced 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.

中文翻译：

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Goldilock 规则、量子元胞自动机和粗粒度

人们可以将某些物理演化视为微观离散模型的紧急有效结果。受经典粗粒度程序的启发，我们提供了一个简单的程序来粗粒度色盲量子细胞自动机，遵循 Goldilock 规则。该过程包括 (i) 将量子细胞自动机 (QCA) 时空分组到大小为 $N$ 的细胞中，(ii) 将细胞状态投影到其边界上，将它们与精细动力学连接起来，这让人联想到边缘- 量子场论中的大量对应；(iii) 描述边境国家的整体动态，我们称之为信号；(iv) 为不同大小的单元格构建粗粒度动态。这个简单的玩具模型的副产品是斯托克斯定律的一般离散类比。此外，我们证明了在时空限制下，自动机收敛到狄拉克自由哈密顿量。我们在此介绍的 QCA 可以通过当今的量子平台实现，例如里德堡阵列、俘获离子和超导量子比特。我们希望我们的研究能够为更深入地了解那些分辨率有限的系统铺平道路。