Quantum cellular automata and categorical dualities of spin chains
Corey Jones, Kylan Schatz, Dominic J. Williamson
TL;DR
This work analyzes how dualities between symmetric local operator algebras on 1D spin chains, encoded by bounded-spread isomorphisms under categorical symmetry, can be extended to quantum cellular automata on the full or edge-restricted algebras. It introduces fusion spin chains $A(\mathcal{E},X)$ and leverages Doplicher-Haag-Roberts (DHR) bimodules to derive a precise extension criterion: a duality $\alpha$ extends to a spatial QCA if and only if $DHR(\alpha)(Z(\mathcal{M}))\cong Z(\mathcal{N})$, and the set of extensions forms a torsor over $Aut(Z(\mathcal{M}))$. The KW duality and group-symmetry cases are used to illustrate obstructions and to obtain an Ind × DHR index that classifies dualities, yielding a unified, operator-algebraic framework connecting categorical inclusions, centers, and QCA. This approach provides a robust path to classifying and constructing spatial implementations, with potential extensions to higher dimensions and more general topological orders.
Abstract
Dualities play a central role in the study of quantum spin chains, providing insight into the structure of quantum phase diagrams and phase transitions. In this work we study categorical dualities, which are defined as bounded-spread isomorphisms between algebras of symmetry-respecting local operators on a spin chain. We consider generalized global symmetries that correspond to unitary fusion categories, which are represented by matrix-product operator algebras. A fundamental question about dualities is whether they can be extended to quantum cellular automata on the larger algebra generated by all local operators that respect the unit matrix-product operator. For conventional global symmetries, which are on-site representations of finite groups, this larger algebra is simply the tensor product of algebras associated to individual spins in the chain. We present a solution to the extension problem using the machinery of Doplicher-Haag-Roberts bimodules. Our solution provides a crisp categorical criterion for when an extension of a duality exists. We show that the set of possible extensions form a torsor over the invertible objects in the relevant symmetry category. As a corollary, we obtain a classification result concerning dualities in the group case.