Non-Clifford quantum cellular automata from invertible topological quantum field theories
Abstract
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.
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BibTeX
@misc{sun2026nonclifford,
title = {{Non-Clifford quantum cellular automata from invertible topological quantum field theories}},
author = {Meng Sun and Zongyuan Wang and Bowen Yang and Nathanan Tantivasadakarn and Yu-An Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.21697},
eprint = {2607.21697},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.21697},
}
RIS
TY - UNPB
TI - Non-Clifford quantum cellular automata from invertible topological quantum field theories
AU - Meng Sun
AU - Zongyuan Wang
AU - Bowen Yang
AU - Nathanan Tantivasadakarn
AU - Yu-An Chen
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2607.21697
UR - https://arxiv.org/abs/2607.21697
AB - 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 ℤ_8×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.
ER -