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 interest. We also include a section leading to the $Ω$ -spectrum for QCA over $C^*$ -algebras with unitary circuits.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 interest. We also include a section leading to the $Ω$-spectrum for QCA over $C^*$-algebras with unitary circuits.

math.ATCross-listedarXiv:2602.16572Feb 19, 2026

Quantum cellular automata and invertible phases of matter

Corey Jones, Nikita Sopenko, Ryan Thorngren

We introduce and study (fermionic and bosonic) invertible quasi-local algebras over uniformly locally finite metric spaces $X$ with infinite-dimensional local von Neumann algebras. We show that the group of Brauer equivalence classes of such algebras is isomorphic to both the group of phases of invertible states and the group of stable equivalence classes of quantum cellular automata over $X\times \mathbb{Z}$ . Using K-theory of the symmetric monoidal category of invertible quasi-local algebras and bounded spread isomorphisms, we propose a definition of an $Ω$ -spectrum of invertible phases as conjectured by Kitaev. We then show that the $c=\frac{1}{2}$ chiral Majorana fermion net and the $(E_{8})_{1}$ conformal net provide Brauer non-trivial invertible quasi-local algebras, thus providing explicit constructions of non-trivial invertible states and quantum cellular automata on $\mathbb{Z}^{2}$ . In addition, we show that the time-slice nets of rational diagonal conformal field theories admit lattice degrees of freedom, which implies the discretization of any holomorphic conformal net is invertible.We introduce and study (fermionic and bosonic) invertible quasi-local algebras over uniformly locally finite metric spaces $X$ with infinite-dimensional local von Neumann algebras. We show that the group of Brauer equivalence classes of such algebras is isomorphic to both the group of phases of invertible states and the group of stable equivalence classes of quantum cellular automata over $X\times \mathbb{Z}$. Using K-theory of the symmetric monoidal category of invertible quasi-local algebras and bounded spread isomorphisms, we propose a definition of an $Ω$-spectrum of invertible phases as conjectured by Kitaev. We then show that the $c=\frac{1}{2}$ chiral Majorana fermion net and the $(E_{8})_{1}$ conformal net provide Brauer non-trivial invertible quasi-local algebras, thus providing explicit constructions of non-trivial invertible states and quantum cellular automata on $\mathbb{Z}^{2}$. In addition, we show that the time-slice nets of rational diagonal conformal field theories admit lattice degrees of freedom, which implies the discretization of any holomorphic conformal net is invertible.

math-phCross-listedarXiv:2608.26456Aug 28, 20260 citations

The Group Structure of Quantum Cellular Automata

Michael Hartley Freedman, Jeongwan Haah, Matthew B. Hastings

arXiv:1910.07998Communications in Mathematical PhysicsJan 25, 202233 citations

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 interest. We also include a section leading to the

Ω

-spectrum for QCA over

C^*

-algebras with unitary circuits.

We introduce and study (fermionic and bosonic) invertible quasi-local algebras over uniformly locally finite metric spaces

X

with infinite-dimensional local von Neumann algebras. We show that the group of Brauer equivalence classes of such algebras is isomorphic to both the group of phases of invertible states and the group of stable equivalence classes of quantum cellular automata over

X\times \mathbb{Z}

. Using K-theory of the symmetric monoidal category of invertible quasi-local algebras and bounded spread isomorphisms, we propose a definition of an

Ω

-spectrum of invertible phases as conjectured by Kitaev. We then show that the

c=\frac{1}{2}

chiral Majorana fermion net and the

(E_{8})_{1}

conformal net provide Brauer non-trivial invertible quasi-local algebras, thus providing explicit constructions of non-trivial invertible states and quantum cellular automata on

\mathbb{Z}^{2}

. In addition, we show that the time-slice nets of rational diagonal conformal field theories admit lattice degrees of freedom, which implies the discretization of any holomorphic conformal net is invertible.