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, ${\pi }_0\mathbf{Q}(X)$ classifies QCA up to quantum circuits and stabilization. Notably, the QCA spaces are related by homotopy equivalences $\mathbf{Q}(\ast )\simeq {\mathrm{\Omega }}^n\mathbf{Q}({\mathbb{Z}}^n)$ for all $n$, which shows that the classification of QCA on Euclidean lattices is given by an $\mathrm{\Omega }$-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 $\mathrm{\Omega }$-spectrum for QCA over $C^{\ast }$-algebras with unitary circuits.