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

TL;DR

This work develops a theory of quantum cellular automata over an arbitrary commutative ring and uses algebraic K-theory to construct a space of quantum cellular automata (QCA) on a given metric space $X$.

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.

Figures (5)

• Figure 1: An example of a single block in a gate. Here we break $\operatorname{Mat}(R^{q_0})$ into its two tensor factors $\operatorname{Mat}(R^a)$ and $\operatorname{Mat}(R^b)$, we then swap out the correspondent tensor components $\operatorname{Mat}(R^a) \subseteq \operatorname{Mat}(R^{q_{-2}})$ and $Mat(R^b) \subseteq \operatorname{Mat}(R^{q_1})$ to perform the automorphism.
• Figure 2: A QCA $\alpha$ such that $b(\alpha) = \frac{A}{B} \in K_0(\operatorname{Az}_{R})$. Here we place an alternating sequence of $A \otimes A' \cong \operatorname{Mat}(R^n)$ and $B \otimes B' \cong \operatorname{Mat}(R^m)$ on $\mathbb{Z}$, and $\alpha$ simultaneously transports the tensor factor $B$ to the left and the tensor factor $A$ to the right. We divide the line between $-1$ and $0$ in the construction of the element $\beta(\alpha)$.
• Figure 3: An illustration of $q^S$ and $\alpha^S$ appearing in the proof of Lemma \ref{['lem::blending']}. The figure illustrates the case where $X$ is a point.
• Figure 4: Figure displaying a quantum spin system on $X \times {\mathbb Z}$ such that applying Definition \ref{['defn::admissible']} would produce the admissible tensor factor $\mathcal{B}$.
• Figure 5: Example of an object in $\textbf{C}'_1(W)$ for $W = {\mathbb Z}$. Over each integer $i \in {\mathbb Z}$ lays a tensor factor of $\mathcal{A}({\mathbb Z}, q_i)$. The dashed ovals indicate the support of these tensor factors.

Theorems & Definitions (151)

• Theorem 1
• Theorem 2
• Corollary 2
• Theorem 3
• Theorem 4: The Algebraic QCA Hypothesis
• Definition 5
• Definition 6
• Remark 7
• Definition 8
• Remark 9: Inverse map