Quantum Cellular Automata on Symmetric Subalgebras
- 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
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 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.
Related
- Thermalization with Gaussian Quantum Cellular Automata
- Fermionic quantum cellular automata in 2d are trivial
- Non-abelian quantum cellular automata: 1+1-dimensional SU(2) Yang--Mills with fermions
- Local Measurement Theory for Quantum Cellular Automata
- Non-Clifford quantum cellular automata from invertible topological quantum field theories
BibTeX
@article{ma2026quantum,
title = {{Quantum Cellular Automata on Symmetric Subalgebras}},
author = {Ruochen Ma and Yabo Li and Meng Cheng},
journal = {Quantum},
year = {2026},
month = jun,
doi = {10.22331/q-2026-06-01-2123},
url = {https://doi.org/10.22331/q-2026-06-01-2123},
note = {CC BY},
}
RIS
TY - JOUR
TI - Quantum Cellular Automata on Symmetric Subalgebras
AU - Ruochen Ma
AU - Yabo Li
AU - Meng Cheng
PY - 2026
DA - 2026/06/01
JO - Quantum
DO - 10.22331/q-2026-06-01-2123
UR - https://doi.org/10.22331/q-2026-06-01-2123
AB - 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.
SN - 2521-327X
ER -