2026/07/23 by Meng Sun, Zongyuan Wang, Bowen Yang +2
#quant-ph #cond-mat.str-el #hep-th #math-ph #math.MP #math.QA
Quantum cellular automata (QCAs) describe locality-preserving quantum dynamics and connect quantum information, many-body physics, and topological quantum field theory (TQFT). Constructing a QCA from a TQFT, however, is challenging. Although a topological action can produce a commuting Hamiltonian realizing the desired ground state, it does not by itself specify an automorphism of the full local operator algebra. In this work, we develop a unified algebraic construction that extends the commuting generators of the Hamiltonian to a complete separator-flipper algebra on the full tensor-product Hilbert space, providing a microscopic definition of the corresponding QCA. In three spatial dimensions, our formalism unifies all previously known QCA constructions associated with the \mathbb Z8×\mathbb Z2 subgroup of the Witt group, including the U(1)2 and U(1)4 QCAs. The same algebraic structure directly yields new infinite families of generalized U(1)2 and U(1)4 non-Clifford QCAs in dimensions d=4k-1. We also reformulate the 4-dimensional w2w3 QCA and use it to develop a general construction of QCAs from TQFTs associated with arbitrary products of Wu classes. This construction includes two infinite families. The first consists of w2nw3m QCAs in dimension d=2n+3m-1, while the second consists of w2w4k-1 QCAs in dimension d=4k. As a contrasting result, we explicitly construct finite-depth quantum circuits for the 5-dimensional w32 and w23 QCAs, thereby proving that they are trivial, in agreement with the cobordism classification. Overall, these results convert invertible TQFTs into microscopic QCAs, provide a scalable route to higher-dimensional constructions beyond the Clifford setting, and open a systematic approach to classifying their stable structures and boundary anomalies.