Clifford quantum cellular automata from topological quantum field theories and invertible subalgebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908925480140800 |
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| author | Sun, Meng Yang, Bowen Wang, Zongyuan Tantivasadakarn, Nathanan Chen, Yu-An |
| author_facet | Sun, Meng Yang, Bowen Wang, Zongyuan Tantivasadakarn, Nathanan Chen, Yu-An |
| contents | 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. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_07099 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Clifford quantum cellular automata from topological quantum field theories and invertible subalgebras Sun, Meng Yang, Bowen Wang, Zongyuan Tantivasadakarn, Nathanan Chen, Yu-An Quantum Algebra Strongly Correlated Electrons High Energy Physics - Theory Mathematical Physics Quantum Physics 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. 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. |
| title | Clifford quantum cellular automata from topological quantum field theories and invertible subalgebras |
| topic | Quantum Algebra Strongly Correlated Electrons High Energy Physics - Theory Mathematical Physics Quantum Physics |
| url | https://arxiv.org/abs/2509.07099 |