Quantum Cellular Automata: The Group, the Space, and the Spectrum
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914618630209536 |
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| author | Ji, Mattie Yang, Bowen |
| author_facet | Ji, Mattie Yang, Bowen |
| contents | 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 interest. We also include a section leading to the $Ω$-spectrum for QCA over $C^*$-algebras with unitary circuits. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_16572 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantum Cellular Automata: The Group, the Space, and the Spectrum Ji, Mattie Yang, Bowen Algebraic Topology Mathematical Physics Operator Algebras Rings and Algebras Quantum Physics Primary: 19D23, 81P45. Secondary: 55P47 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 interest. We also include a section leading to the $Ω$-spectrum for QCA over $C^*$-algebras with unitary circuits. |
| title | Quantum Cellular Automata: The Group, the Space, and the Spectrum |
| topic | Algebraic Topology Mathematical Physics Operator Algebras Rings and Algebras Quantum Physics Primary: 19D23, 81P45. Secondary: 55P47 |
| url | https://arxiv.org/abs/2602.16572 |