Non-Clifford quantum cellular automata from invertible topological quantum field theories

Abstract

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 $\mathbb Z_8\times\mathbb Z_2$ 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 $w_2w_3$ 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 $w_2^nw_3^m$ QCAs in dimension $d=2n+3m-1$, while the second consists of $w_2w_{4k-1}$ QCAs in dimension $d=4k$. As a contrasting result, we explicitly construct finite-depth quantum circuits for the 5-dimensional $w_3^2$ and $w_2^3$ 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.