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.
1. Introduction 1.1. Quantum Cellular Automata. Quantum cellular automata (QCA) originate in quantum many-body physics and quantum information as locality-preserving automorphisms of the C * -algebras associated to quantum spin systems. They provide a mathematically precise framework for studying reversible dynamics constrained by the large-scale geometry of the system.
Among the many problems surrounding QCA, the classification problem has attracted the most attention. Early progress resulted in the complete classification in one dimension [Gro+12]. What followed was a period of rapid progress. We mention only a few of the many developments. The group-theoretic foundation was laid out [FH20;FHH22]. Highly nontrivial examples were constructed [HFH23;CH23;Shi+22]. A special class of QCA, known as Clifford QCA, was classified using algebraic L-groups [Haa25;Yan25]. Connections with topological field theory have begun to emerge [FHH24;Sun+25], and a bulk-boundary correspondence was established and related to commuting Hamiltonian models [RY25;Haa23;Haa21]. Last but not least, exciting connections to lattice anomalies and invertible states have been discovered [KS25b; Fen+25; KS25a; KX25; CGT25; TLE25].
Perhaps the most surprising aspect of the classification problem, and what drew the authors’ attention, is its connection to homotopy theory. An early hint of this connection appeared in an appendix of a paper [Fre+22] on manifold topology, but it profoundly influenced our perspective. Several proposals for the homotopy type of QCA have appeared in the physics literature, including formulations in terms of crossed n-cubes [KS25a; KX25] and in the language of ∞-categories [CGT25; TLE25]. There is now a folklore conjecture, which we refer to as the QCA conjecture: underlying the classification of QCA is an Ω-spectrum indexed by the spatial dimension of the lattice Z n . These developments share a common feature: the techniques involved are largely algebraic and, in many cases, do not rely on the choice of base field C or on the C * -algebraic structure. Therefore, we raise the question of whether one can develop a theory of “QCA” over an arbitrary commutative base ring. Moreover, does there exist, for each such ring, an associated Ω-spectrum governing the classification of QCA’s? We refer to this statement as the algebraic QCA hypothesis.
In this paper, we develop the theory from first principles and prove the algebraic QCA hypothesis. In doing so, we construct a family of Ω-spectra, one for each commutative ring. More specifically, for each ring R we have a sequence of infinite loop spaces:
(1.1) Q( * ) ≃ ΩQ(Z 1 ), Q(Z 1 ) ≃ ΩQ(Z 2 ), …, Q(Z n-1 ) ≃ ΩQ(Z n ), … .
We prove that Q(Z 1 ) coincides with a space known as the algebraic K-theory of Azumaya algebras over R [Wei81a], but little is known about the delooping Q(Z n ) for general n.
It is conjectured, supported by arguments using topological phases of matter [Haa21;FHH24;Sun+25], that when R = C, the group 1 π 0 Q(Z 3 ) is equal or ‘close to’ the Witt group of braided fusion C-linear categories [Dav+10; DNO13].
1 More precisely, the conjecture applies to the original formulation in physics which we discuss in Section 5.
This leads to another question: over a field R = k, does the group π 0 Q(Z 3 ) reproduce the Witt group of braided fusion k-linear categories? Part of our motivation for developing the theory over general rings is to eventually understand what QCA detects about the ring and to answer these questions.
1.2. Algebraic K-Theory and Group Completion. Over a ring R, the strategy we use to construct the QCA spaces Q(Z n ) is via algebraic K-theory and group completion. Algebraic K-theory, on a high level, is the study of how to break and assemble objects apart linearly, which makes them amenable to classification questions.
Given a ring R, its K 0 -group is defined as the algebraic group completion of a certain commutative monoid associated to R, and K 0 (R) is used to classify finitely generated projective modules over R. Quillen defined models for higher K-groups [Qui72;Qui73;Gra76] based on the principle that the higher K-theories K i (R) of R should be the homotopy groups π i (K(R)) of a space K(R), with K 0 (R) = π 0 (K(R)). Segal [Seg74] constructed a model for K(R) as the topological group completion of a topological/homotopical version of a monoid. In particular, Segal’s construction applies to any symmetric monoidal category C and produces a corresponding K-theory space K(C).
Instead of going up, Bass [Bas68] defined the lower K-groups (also known as negative K-groups) K -i (R) of a ring R. Pederson showed [Ped84] that K -i (R) can be recovered as π 1 K(C i+1 (R)), where C i+1 (R) is a symmetric monoidal category associated to R. To give an informal description of this category C i+1 (R) (see [Ped84;PW85]), an object in C i+1 (R) is created by placing a finitely generated free R-module on each point of the lattice Z i+1 ⊂ R i+
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