Quantum Cellular Automata: The Group, the Space, and the Spectrum

• URL: http://arxiv.org/abs/2602.16572v1
• Date: Wed, 18 Feb 2026 16:15:36 GMT
• Title: Quantum Cellular Automata: The Group, the Space, and the Spectrum
• Authors: Mattie Ji, Bowen Yang,
• Abstract summary: We construct a space $mathbfQ(X)$ of quantum cellular automata on a given metric space $X$.<n>The QCA spaces are related by homotopy equivalences $mathbfQ simeq n mathbfQ(mathbbZn)$ for all $n$, which shows that the classification of QCA on Euclidean lattices is given by an $$-spectrum indexed by the dimension $n$.
• Score: 8.162672407534899
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Over an arbitrary commutative ring $R$, we develop a theory of quantum cellular automata. We then use algebraic K-theory to construct a space $\mathbf{Q}(X)$ of quantum cellular automata (QCA) on a given metric space $X$. In most cases of interest, $π_0 \mathbf{Q}(X)$ classifies QCA up to quantum circuits and stabilization. Notably, the QCA spaces are related by homotopy equivalences $\mathbf{Q}(*) \simeq Ω^n \mathbf{Q}(\mathbb{Z}^n)$ for all $n$, which shows that the classification of QCA on Euclidean lattices is given by an $Ω$-spectrum indexed by the dimension $n$. As a corollary, we also obtain a non-connective delooping of the K-theory of Azumaya $R$-algebras, which may be of independent interests. We also include a section leading to the $Ω$-spectrum for QCA over $C^*$-algebras with unitary circuits.

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