Quantum cellular automata (QCA) are discrete models of space and time homogeneous quantum field theories (QFTs) and regarded as natural candidates for quantum simulation. Description of a QCA over the separable Hilbert space of finite, unbounded configurations (UFC Hilbert space) with unitary state evolution is the {\it Schrödinger template}, and over the incomplete infinite tensor product algebra (ITPA) with evolution by algebra automorphism is the {\it Heisenberg template}. Whether every Heisenberg template admits an equivalent Schrödinger template is a foundational question, and one that has persisted as an open problem. In the present paper we prove that for every Heisenberg template an equivalent Schrödinger template exists. We frame the question from a representation theory standpoint, using constructs and results from the representation theory of finite and countably infinite dimensional vector spaces and from category theory to answer it. With the previously known existence of a Heisenberg template for every Schrödinger template, our result establishes the equivalence of the templates.
Axiomatic definitions of Quantum Cellular Automata (QCA) are foundational in discrete space-time models of physics. Templates of axiomatic QCA serve as primary points of investigations concerning properties, structural results, and classes of QCA that are potentially significant for quantum algorithms and simulations. We examine, from a representation theory standpoint, the relation between the templates corresponding to the Schr\{o}dinger and the Heisenberg pictures of evolution for QCA over finite, unbounded configurations.
Quantum cellular automata (QCA) are models of quantum computation of particular interest from the point of view of quantum simulation. Quantum lattice gas automata (QLGA - equivalently partitioned quantum cellular automata) represent an interesting subclass of QCA. QLGA have been more deeply analyzed than QCA, whereas general QCA are likely to capture a wider range of quantum behavior. Discriminating between QLGA and QCA is therefore an important question. In spite of much prior work, classifying which QCA are QLGA has remained an open problem. In the present paper we establish necessary and sufficient conditions for unbounded, finite QCA (finitely many active cells in a quiescent background) to be QLGA. We define a local condition that classifies those QCA that are QLGA, and we show that there are QCA that are not QLGA. We use a number of tools from functional analysis of separable Hilbert spaces and representation theory of associative algebras that enable us to treat QCA on finite but unbounded configurations in full detail.
Quantum cellular automata (QCA) constitute space and time homogeneous discrete models for quantum field theories (QFTs). Although QFTs are defined without reference to particles, computations are done in terms of Feynman diagrams, which are explicitly interpreted in terms of interacting particles. Similarly, the easiest QCA to construct are quantum lattice gas automata (QLGA). A natural question then is, which QCA are not QLGA? Here we construct a nontrivial example of such a QCA; it provides a simple model in
1+1 dimensions with no particle interpretation at the scale where the QCA dynamics are homogeneous.
Over an arbitrary commutative ring
R, we develop a theory of quantum cellular automata. We then use algebraic K-theory to construct a space
Q(X) of quantum cellular automata (QCA) on a given metric space
X. In most cases of interest,
π0Q(X) classifies QCA up to quantum circuits and stabilization. Notably, the QCA spaces are related by homotopy equivalences
Q(∗)≃ΩnQ(Zn) 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 define and study quantum cellular automata (QCA). We show that they are reversible and that the neighborhood of the inverse is the opposite of the neighborhood. We also show that QCA always admit, modulo shifts, a two-layered block representation. Note that the same two-layered block representation result applies also over infinite configurations, as was previously shown for one-dimensional systems in the more elaborate formalism of operators algebras [18]. Here the proof is simpler and self-contained, moreover we discuss a counterexample QCA in higher dimensions.
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
Z8×Z2 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
w2w3 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:
w2nw3m QCAs in dimension
d=2n+3m−1 and
w2w4k−1 QCAs in dimension
d=4k. For the 5-dimensional
w32 and
w23 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.