Quantum cellular automata (QCA) are reviewed, including early and more recent proposals. QCA are a generalization of (classical) cellular automata (CA) and in particular of reversible CA. The latter are reviewed shortly. An overview is given over early attempts by various authors to define one-dimensional QCA. These turned out to have serious shortcomings which are discussed as well. Various proposals subsequently put forward by a number of authors for a general definition of one- and higher-dimensional QCA are reviewed and their properties such as universality and reversibility are discussed.Quantum cellular automata (QCA) are reviewed, including early and more recent proposals. QCA are a generalization of (classical) cellular automata (CA) and in particular of reversible CA. The latter are reviewed shortly. An overview is given over early attempts by various authors to define one-dimensional QCA. These turned out to have serious shortcomings which are discussed as well. Various proposals subsequently put forward by a number of authors for a general definition of one- and higher-dimensional QCA are reviewed and their properties such as universality and reversibility are discussed.

arXiv:0808.0679Aug 5, 20081 citation

Quantum Cellular Automata

Massimo Macucci

IMPERIAL COLLEGE PRESS eBooksMar 1, 200632 citations

Quantum Cellular Automata

Karoline Wiesner

Springer eBooksNov 2, 20115 citations

Time asymptotics and entanglement generation of Clifford quantum cellular automata

Johannes Gütschow, Sonja Uphoff, Reinhard F. Werner +1 more

We consider Clifford quantum cellular automata (CQCAs) and their time-evolution. CQCAs are an especially simple type of quantum cellular automata, yet they show complex asymptotics and can even be a basic ingredient for universal quantum computation. In this work we study the time evolution of different classes of CQCAs. We distinguish between periodic CQCAs, fractal CQCAs, and CQCAs with gliders. We then identify invariant states and study convergence properties of classes of states, such as quasifree and stabilizer states. Finally, we consider the generation of entanglement analytically and numerically for stabilizer and quasifree states.We consider Clifford quantum cellular automata (CQCAs) and their time-evolution. CQCAs are an especially simple type of quantum cellular automata, yet they show complex asymptotics and can even be a basic ingredient for universal quantum computation. In this work we study the time evolution of different classes of CQCAs. We distinguish between periodic CQCAs, fractal CQCAs, and CQCAs with gliders. We then identify invariant states and study convergence properties of classes of states, such as quasifree and stabilizer states. Finally, we consider the generation of entanglement analytically and numerically for stabilizer and quasifree states.

arXiv:0906.3195Journal of Mathematical PhysicsJan 1, 201053 citations

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

Mattie Ji, Bowen Yang

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

Karoline Wiesner

Springer eBooksJan 1, 20185 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.