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

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, π₀ mathbf{Q}(X) classifies QCA up to quantum circuits and stabilization. Notably, the QCA spaces are related by homotopy equivalences mathbf{Q}(*) simeq Ωⁿ 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.