Quantum Cellular Automata on Symmetric Subalgebras

Ruochen Ma; Yabo Li; Meng Cheng

doi:10.22331/q-2026-06-01-2123

Quantum Cellular Automata on Symmetric Subalgebras

Ruochen Ma¹, Yabo Li², and Meng Cheng³

¹Kadanoff Center for Theoretical Physics & Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, USA
²Center for Quantum Phenomena, Department of Physics, New York University, 726 Broadway, New York, New York 10003, USA
³Department of Physics, Yale University, New Haven, Connecticut 06511-8499, USA

Published:	2026-06-01, volume 10, page 2123
Editor:	Álvaro Alhambra
Eprint:	arXiv:2411.19280v3
Doi:	https://doi.org/10.22331/q-2026-06-01-2123
Citation:	Quantum 10, 2123 (2026).

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Abstract

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 $\mathbb{Z}_2$ symmetric subalgebra, demonstrating that it maps to the $e$-$m$ permutation in the two-dimensional toric code and has an irrational index of $\sqrt{2}$. Therefore, it cannot be extended to a QCA over the full local operator algebra and mixes nontrivially with lattice translations.

Featured image: A one-dimensional quantum cellular automaton may act only on the symmetry-invariant subalgebra of local operators. Such subalgebra automata are classified by two invariants: the induced permutation of anyons in a two-dimensional $G$ gauge theory, and a generalized flow index $\mathrm{ind}(\alpha)$. Kramers–Wannier duality is the basic example: it exchanges $e$ and $m$ anyons and has index $\sqrt{2}$, showing that it cannot be extended to an ordinary automaton of the full local algebra.

Popular summary

Quantum cellular automata are idealized time evolutions in which quantum information can move only a bounded distance in one time step. In one dimension, ordinary quantum cellular automata are well understood: up to finite-depth local circuits, they are classified by an index that measures the net flow of quantum information. Many important transformations in quantum matter, however, are not ordinary automata on the full set of local operators. They act naturally only on the operators that respect a symmetry. The Kramers–Wannier duality of the Ising chain is the simplest example: it is local and invertible on the $\mathbb{Z}_2$-symmetric operators, but it does not preserve symmetry-charged local operators.

This work develops a classification of such symmetry-restricted automata for finite Abelian symmetries. We show that two pieces of data completely determine them, up to symmetric finite-depth circuits. The first is an anyon permutation in a two-dimensional gauge theory associated with the symmetry. The second is a generalized information-flow index for the symmetric subalgebra. Together these invariants reveal phenomena that cannot occur for ordinary quantum cellular automata. For example, Kramers–Wannier duality exchanges the electric and magnetic anyons of the toric code and has index $\sqrt{2}$, so it cannot be extended to an ordinary automaton on the full local algebra.

This perspective gives a systematic lattice framework for non-invertible symmetries and for their possible mixing with spatial translations.

► BibTeX data

@article{Ma2026quantumcellular,
  doi = {10.22331/q-2026-06-01-2123},
  url = {https://doi.org/10.22331/q-2026-06-01-2123},
  title = {Quantum {C}ellular {A}utomata on {S}ymmetric {S}ubalgebras},
  author = {Ma, Ruochen and Li, Yabo and Cheng, Meng},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2123},
  month = jun,
  year = {2026}
}

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The above citations are from SAO/NASA ADS (last updated successfully 2026-08-19 16:30:23). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2026-08-19 16:30:16).

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