Clifford Quantum Cellular Automata from Topological Quantum Field Theories and Invertible Subalgebras
Meng Sun, Bowen Yang, Zongyuan Wang, Nathanan Tantivasadakarn, Yu-An Chen · PRX Quantum · 2026
We present a general framework for constructing quantum cellular automata (QCAs) from topological quantum field theories (TQFTs) and invertible subalgebras (ISAs) using the cup-product formalism. This approach explicitly realizes all Z 2 and 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 (FDQCs) and lattice translations. In particular, we demonstrate that the Z 2 Clifford QCAs in ( 4 l + 1 ) spatial dimensions can be disentangled by non-Clifford FDQCs. Our construction applies beyond cubic lattices, allowing Z 2 QCAs to be defined on arbitrary cellulations. Furthermore, we explicitly construct invertible subalgebras in higher dimensions, obtaining Z 2 ISAs in 2 l spatial dimensions and Z p ISAs in ( 4 l − 2 ) spatial dimensions. These ISAs give rise to Z 2 QCAs in ( 2 l + 1 ) dimensions and Z p QCAs in ( 4 l − 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.