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arXiv:2603.08702

Approximate QCAs in one dimension using approximate algebras

Daniel Ranard, Michael Walter, Freek Witteveen

quant-phmath-phSubmitted Mar 9, 2026• Announced Mar 10, 2026• 0 citations
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Abstract

Quantum cellular automata (QCAs) are automorphisms of tensor product algebras that preserve locality, with local quantum circuits as a simple example. We study approximate QCAs, where the locality condition is only satisfied up to a small error, as occurs for local quantum dynamics on the lattice. A priori, approximate QCAs could exhibit genuinely new behavior, failing to be well-approximated by any exact QCA. We show this does not occur in one dimension: every approximate QCA on a finite circle can be rounded to a strict QCA with approximately the same action on local operators, so these systems are classified by the same index as in the exact case. Previous work considered the case of the infinite line, by using global methods not amenable to finite systems. Our new approach proceeds locally and now applies to finite systems, including circles or homomorphisms from sub-intervals. We extract exact local boundary algebras from the approximate QCA restricted to local patches, then glue these to form a strict QCA. The key technical ingredient is a robust notion of the intersection of two subalgebras: when the projections onto two subalgebras approximately commute, we construct an exact subalgebra that serves as a stable proxy for their intersection. This construction uses a recent theorem of Kitaev on the rigidity of approximate $C^*$-algebras.

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19 pages

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

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Quantum cellular automata (QCAs) are automorphisms of tensor product algebras that preserve locality, with local quantum circuits as a simple example. We study approximate QCAs, where the locality condition is only satisfied up to a small error, as occurs for local quantum dynamics on the lattice. A priori, approximate QCAs could exhibit genuinely new behavior, failing to be well-approximated by any exact QCA. We show this does not occur in one dimension: every approximate QCA on a finite circle can be rounded to a strict QCA with approximately the same action on local operators, so these systems are classified by the same index as in the exact case. Previous work considered the case of the infinite line, by using global methods not amenable to finite systems. Our new approach proceeds locally and now applies to finite systems, including circles or homomorphisms from sub-intervals. We extract exact local boundary algebras from the approximate QCA restricted to local patches, then glue these to form a strict QCA. The key technical ingredient is a robust notion of the intersection of two subalgebras: when the projections onto two subalgebras approximately commute, we construct an exact subalgebra that serves as a stable proxy for their intersection. This construction uses a recent theorem of Kitaev on the rigidity of approximate C∗C^*C∗-algebras.
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\mathbb{Z}_8\times\mathbb{Z}_2Z8​×Z2​ subgroup of the Witt group, including the U(1)2U(1)_2U(1)2​ and U(1)4U(1)_4U(1)4​ QCAs. The same algebraic structure yields new infinite families of generalized U(1)2U(1)_2U(1)2​ and U(1)4U(1)_4U(1)4​ non-Clifford QCAs in spatial dimensions d=4k−1d=4k-1d=4k−1. We also reformulate the 4-dimensional w2w3w_2w_3w2​w3​ 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: w2nw3mw_2^nw_3^mw2n​w3m​ QCAs in dimension d=2n+3m−1d=2n+3m-1d=2n+3m−1 and w2w4k−1w_2w_{4k-1}w2​w4k−1​ QCAs in dimension d=4kd=4kd=4k. For the 5-dimensional w32w_3^2w32​ and w23w_2^3w23​ 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.
Over an arbitrary commutative ring RRR, we develop a theory of quantum cellular automata. We then use algebraic K-theory to construct a space Q(X)\mathbf{Q}(X)Q(X) of quantum cellular automata (QCA) on a given metric space XXX. In most cases of interest, π0Q(X)π_0 \mathbf{Q}(X)π0​Q(X) classifies QCA up to quantum circuits and stabilization. Notably, the QCA spaces are related by homotopy equivalences Q(∗)≃ΩnQ(Zn)\mathbf{Q}(*) \simeq Ω^n \mathbf{Q}(\mathbb{Z}^n)Q(∗)≃ΩnQ(Zn) for all nnn, which shows that the classification of QCA on Euclidean lattices is given by an ΩΩΩ-spectrum indexed by the dimension nnn. As a corollary, we also obtain a non-connective delooping of the K-theory of Azumaya RRR-algebras, which may be of independent interest. We also include a section leading to the ΩΩΩ-spectrum for QCA over C∗C^*C∗-algebras with unitary circuits.
Fermionic quantum cellular automata (QCA) are automorphisms of local fermionic operator algebras (Z2\mathbb Z_2Z2​-graded superalgebras) that have bounded spread: they map local operators to nearby operators. We prove that every 2-dimensional fermionic QCA on a locally finite-dimensional algebra is a composition of local automorphisms and a fermionic shift. This is implied by our result that every locally finite-dimensional fermionic invertible subalgebra in a one-dimensional lattice is Brauer trivial, \textit{i.e.}, it is stably bounded-spread isomorphic to a tensor product fermionic algebra.
We investigate quantum cellular automata (QCA) on one-dimensional spin systems defined over a subalgebra of the full local operator algebra – the symmetric subalgebra under a finite Abelian group symmetry G . For systems where each site carries a regular representation of G , we establish a complete classification of such subalgebra QCAs based on two topological invariants: (1) a surjective homomorphism from the group of subalgebra QCAs to the group of anyon permutation symmetries in a ( 2 + 1 ) d G gauge theory; and (2) a generalization of the Gross-Nesme-Vogts-Werner (GNVW) index that characterizes the flow of the symmetric subalgebra. Specifically, two subalgebra QCAs correspond to the same anyon permutation and share the same index if and only if they differ by a finite-depth unitary circuit composed of G -symmetric local gates. We also identify a set of operations that generate all subalgebra QCAs through finite compositions. As an example, we examine the Kramers-Wannier duality on a Z 2 symmetric subalgebra, demonstrating that it maps to the e - m permutation in the two-dimensional toric code and has an irrational index of 2 . Therefore, it cannot be extended to a QCA over the full local operator algebra and mixes nontrivially with lattice translations.
Discretizing spacetime is often a natural step towards modelling physical systems. For quantum systems, if we also demand a strict bound on the speed of information propagation, we get quantum cellular automata (QCAs). These originally arose as an alternative paradigm for quantum computation, though more recently they have found application in understanding topological phases of matter and have} been proposed as models of periodically driven (Floquet) quantum systems, where QCA methods were used to classify their phases. QCAs have also been used as a natural discretization of quantum field theory, and some interesting examples of QCAs have been introduced that become interacting quantum field theories in the continuum limit. This review discusses all of these applications, as well as some other interesting results on the structure of quantum cellular automata, including the tensor-network unitary approach, the index theory and higher dimensional classifications of QCAs.
We present a general framework for constructing quantum cellular automata (QCAs) from topological quantum field theories (TQFTs) and invertible subalgebras (ISAs) using the cup-product formalism. This approach explicitly realizes all Z 2 and Z p Clifford QCAs (for prime p ) in all admissible dimensions, in precise agreement with the classification predicted by algebraic L -theory. We determine the orders of these QCAs by explicitly showing that finite powers reduce to the identity up to finite-depth quantum circuits (FDQCs) and lattice translations. In particular, we demonstrate that the Z 2 Clifford QCAs in ( 4 l + 1 ) spatial dimensions can be disentangled by non-Clifford FDQCs. Our construction applies beyond cubic lattices, allowing Z 2 QCAs to be defined on arbitrary cellulations. Furthermore, we explicitly construct invertible subalgebras in higher dimensions, obtaining Z 2 ISAs in 2 l spatial dimensions and Z p ISAs in ( 4 l − 2 ) spatial dimensions. These ISAs give rise to Z 2 QCAs in ( 2 l + 1 ) dimensions and Z p QCAs in ( 4 l − 1 ) dimensions. We further prove that the QCAs in 3 spatial dimensions constructed via TQFTs and ISAs are equivalent by identifying their boundary algebras and show that this approach extends to higher dimensions. Together, these results establish a unified and dimension-periodic framework for Clifford QCAs, connecting their explicit lattice realizations to field theories.