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

Quantum Cellular Automata: Theory and Applications

Pérez Delgado, Carlos Alberto Conceição António

This thesis presents a model of Quantum Cellular Automata (QCA). The presented formalism is a natural quantization of the classical Cellular Automata (CA). It is based on a lattice of qudits, and an update rule consisting of local unitary operators that commute with their own lattice translations. One purpose of this model is to act as a theoretical model of quantum computation, similar to the quantum circuit model. The main advantage that QCA have over quantum circuits is that QCA make considerably fewer demands on the underlying hardware. In particular, as opposed to direct implementations of quantum circuits, the global evolution of the lattice in the QCA model does not assume independent control over individual \emph{qudits}. Rather, all qudits are to be addressed collectively in parallel. The QCA model is also shown to be an appropriate abstraction for space-homogeneous quantum phenomena, such as quantum lattice gases, spin chains and others. Some results that show the benefits of basing the model on local unitary operators are shown: computational universality, strong connections to the circuit model, simple implementation on quantum hardware, and a series of applications. A detailed discussion will be given on one particular application of QCA that lies outside either computation or simulation: single-spin measurement. This algorithm uses the techniques developed in this thesis to achieve a result normally considered hard in physics. It serves well as an example of why QCA are interesting in their own right.This thesis presents a model of Quantum Cellular Automata (QCA). The presented formalism is a natural quantization of the classical Cellular Automata (CA). It is based on a lattice of qudits, and an update rule consisting of local unitary operators that commute with their own lattice translations. One purpose of this model is to act as a theoretical model of quantum computation, similar to the quantum circuit model. The main advantage that QCA have over quantum circuits is that QCA make considerably fewer demands on the underlying hardware. In particular, as opposed to direct implementations of quantum circuits, the global evolution of the lattice in the QCA model does not assume independent control over individual \emph{qudits}. Rather, all qudits are to be addressed collectively in parallel. The QCA model is also shown to be an appropriate abstraction for space-homogeneous quantum phenomena, such as quantum lattice gases, spin chains and others. Some results that show the benefits of basing the model on local unitary operators are shown: computational universality, strong connections to the circuit model, simple implementation on quantum hardware, and a series of applications. A detailed discussion will be given on one particular application of QCA that lies outside either computation or simulation: single-spin measurement. This algorithm uses the techniques developed in this thesis to achieve a result normally considered hard in physics. It serves well as an example of why QCA are interesting in their own right.

Sep 26, 20074 citations

Quantum Cellular Automata

Ioannis Liliopoulos, Georgios D. Varsamis, Ioannis G. Karafyllidis

Emergence, complexity and computationJan 1, 20251 citation

Non-Clifford quantum cellular automata from invertible topological quantum field theories

Meng Sun, Zongyuan Wang, Bowen Yang +2 more

Quantum cellular automata (QCAs) describe locality-preserving quantum dynamics and link quantum information, many-body physics, and topological quantum field theory (TQFT). Constructing a QCA from a TQFT, however, remains challenging: although a topological action yields a Hamiltonian with commuting terms that realizes the desired ground state, it does not by itself specify an automorphism of the full operator algebra. We develop a unified algebraic construction that extends such Hamiltonians to complete separator-flipper algebras on the full tensor-product Hilbert space, thereby defining the corresponding QCAs microscopically. In 3 spatial dimensions, our framework unifies all previously known QCA constructions associated with the $\mathbb{Z}_8\times\mathbb{Z}_2$ subgroup of the Witt group, including the $U(1)_2$ and $U(1)_4$ QCAs. The same algebraic structure yields new infinite families of generalized $U(1)_2$ and $U(1)_4$ non-Clifford QCAs in spatial dimensions $d=4k-1$ . We also reformulate the 4-dimensional $w_2w_3$ QCA and use it to construct QCAs from TQFTs associated with arbitrary products of Stiefel--Whitney classes. For example, we explicitly construct two infinite families: $w_2^nw_3^m$ QCAs in dimension $d=2n+3m-1$ and $w_2w_{4k-1}$ QCAs in dimension $d=4k$ . For the 5-dimensional $w_3^2$ and $w_2^3$ QCAs, our construction gives finite-depth quantum circuits, consistent with the cobordism classification. More generally, we show that our construction yields a finite-depth quantum circuit whenever the corresponding Stiefel-Whitney number vanishes on every closed oriented manifold. Overall, these results convert invertible TQFTs into microscopic QCAs, provide higher-dimensional constructions beyond the Clifford setting, and open a systematic route to characterizing their stable structures and boundary anomalies.Quantum cellular automata (QCAs) describe locality-preserving quantum dynamics and link quantum information, many-body physics, and topological quantum field theory (TQFT). Constructing a QCA from a TQFT, however, remains challenging: although a topological action yields a Hamiltonian with commuting terms that realizes the desired ground state, it does not by itself specify an automorphism of the full operator algebra. We develop a unified algebraic construction that extends such Hamiltonians to complete separator-flipper algebras on the full tensor-product Hilbert space, thereby defining the corresponding QCAs microscopically. In 3 spatial dimensions, our framework unifies all previously known QCA constructions associated with the $\mathbb{Z}_8\times\mathbb{Z}_2$ subgroup of the Witt group, including the $U(1)_2$ and $U(1)_4$ QCAs. The same algebraic structure yields new infinite families of generalized $U(1)_2$ and $U(1)_4$ non-Clifford QCAs in spatial dimensions $d=4k-1$. We also reformulate the 4-dimensional $w_2w_3$ QCA and use it to construct QCAs from TQFTs associated with arbitrary products of Stiefel--Whitney classes. For example, we explicitly construct two infinite families: $w_2^nw_3^m$ QCAs in dimension $d=2n+3m-1$ and $w_2w_{4k-1}$ QCAs in dimension $d=4k$. For the 5-dimensional $w_3^2$ and $w_2^3$ QCAs, our construction gives finite-depth quantum circuits, consistent with the cobordism classification. More generally, we show that our construction yields a finite-depth quantum circuit whenever the corresponding Stiefel-Whitney number vanishes on every closed oriented manifold. Overall, these results convert invertible TQFTs into microscopic QCAs, provide higher-dimensional constructions beyond the Clifford setting, and open a systematic route to characterizing their stable structures and boundary anomalies.

quant-pharXiv:2607.21697Jul 27, 20260 citations

Quantum Cellular Automata

Karoline Wiesner

Springer eBooksNov 2, 20115 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.

\mathbb{Z}_8\times\mathbb{Z}_2

subgroup of the Witt group, including the

U(1)_2

and

U(1)_4

QCAs. The same algebraic structure yields new infinite families of generalized

U(1)_2

and

U(1)_4

non-Clifford QCAs in spatial dimensions

d=4k-1

. We also reformulate the 4-dimensional

w_2w_3

QCA and use it to construct QCAs from TQFTs associated with arbitrary products of Stiefel--Whitney classes. For example, we explicitly construct two infinite families:

w_2^nw_3^m

QCAs in dimension

d=2n+3m-1

and

w_2w_{4k-1}

QCAs in dimension

d=4k

. For the 5-dimensional

w_3^2

and

w_2^3

QCAs, our construction gives finite-depth quantum circuits, consistent with the cobordism classification. More generally, we show that our construction yields a finite-depth quantum circuit whenever the corresponding Stiefel-Whitney number vanishes on every closed oriented manifold. Overall, these results convert invertible TQFTs into microscopic QCAs, provide higher-dimensional constructions beyond the Clifford setting, and open a systematic route to characterizing their stable structures and boundary anomalies.