Clifford quantum cellular automata from topological quantum field theories and invertible subalgebras

• URL: http://arxiv.org/abs/2509.07099v1
• Date: Mon, 08 Sep 2025 18:00:13 GMT
• Title: Clifford quantum cellular automata from topological quantum field theories and invertible subalgebras
• Authors: Meng Sun, Bowen Yang, Zongyuan Wang, Nathanan Tantivasadakarn, Yu-An Chen,
• Abstract summary: We present a framework for constructing quantum cellular automata (QCA) from topological quantum field theories (TQFT) and invertible subalgebras (ISA)<n>This approach explicitly realizes all $mathbbZ$ and $mathbbZ_p$ Clifford QCAs in all admissible dimensions.
• Score: 10.600087006666332
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We present a general framework for constructing quantum cellular automata (QCA) from topological quantum field theories (TQFT) and invertible subalgebras (ISA) using the cup-product formalism. This approach explicitly realizes all $\mathbb{Z}_2$ and $\mathbb{Z}_p$ 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 (FDQC) and lattice translations. In particular, we demonstrate that the $\mathbb{Z}_2$ Clifford QCAs in $(4l+1)$ spatial dimensions can be disentangled by non-Clifford FDQCs. Our construction applies beyond cubic lattices, allowing $\mathbb{Z}_2$ QCAs to be defined on arbitrary cellulations. Furthermore, we explicitly construct invertible subalgebras in higher dimensions, obtaining $\mathbb{Z}_2$ ISAs in $2l$ spatial dimensions and $\mathbb{Z}_p$ ISAs in $(4l-2)$ spatial dimensions. These ISAs give rise to $\mathbb{Z}_2$ QCAs in $(2l+1)$ dimensions and $\mathbb{Z}_p$ QCAs in $(4l-1)$ dimensions. Together, these results establish a unified and dimension-periodic framework for Clifford QCAs, connecting their explicit lattice realizations to field theories.

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