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

TL;DR

This work constructs and classifies Clifford QCAs in arbitrary dimensions by two complementary routes: discretized TQFT actions via cup products and invertible subalgebras (ISAs). It shows that all $ t Z_2$ and $ t Z_p$ Clifford QCAs (prime $p$) arise in a dimension-periodic pattern consistent with algebraic L-theory, and it determines their orders by reducing powers to FDQCs and lattice translations. It proves the equivalence of QCAs built from TQFTs and ISAs through boundary-algebra analysis, and develops an algebraic K-theory framework to understand when two QCAs are genuinely distinct. The results unify lattice realizations with field-theoretic data, yield explicit higher-dimensional constructions on arbitrary cellulations, and provide a foundation for exploring non-Clifford QCAs and higher-form symmetries in quantum dynamics.

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. 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.

Figures (11)

• Figure 1: Examples of polynomial expressions for Pauli operators adapted from Ref. liang2023extracting. The flux operator on a plaquette and the $XZ$ operator on edges are shown. Red edges mark the chosen base edge, while all other edges follow from it by lattice translations. Monomials such as $x^2 y^2$ and $x^2$ specify operator locations relative to the origin. A detailed review is provided in Appendix \ref{['app: Review of the Laurent polynomial formalism']}.
• Figure 2: An illustration on the dual lattice showing the construction of the flux terms $B_e$ via the action of the coboundary matrix ${\sansmath{\bm \delta}}_{f,\delta e}$ on the vector of operators $X_f$. A Pauli operator $X_f$ is supported on each face $f$, and the product $X_f {\sansmath{\bm \delta}}_{f,\delta e}$ yields the flux term $X_{\delta e} = B_e$. The three colored plaquettes correspond to the $yz$, $xy$, and $xy$ orientations, with the associated polynomial expressions shown below.
• Figure 3: The flippers $\overline{X}_f^A$ and $\overline{X}_f^B$ of the 3-fermion QCA on the dual lattice, with $G^A$ and $G^B$ defined in Eq. \ref{['eq: definition of G']}.
• Figure 4: A valid choice of parent Hamiltonian for the gauged 1-form SPT phase associated with the cocycle $S = \tfrac{k}{p} B_2 \cup B_2 \in H^4(B^2\mathbb{Z}_p, \mathbb{R}/\mathbb{Z})$, illustrated on the dual lattice. However, the Hamiltonian terms are not independent, and therefore cannot be used directly as separators in the QCA construction.
• Figure 5: Separators $\overline{Z}_f^{2k}$ shown on the dual lattice.

Theorems & Definitions (20)

• Definition 1: Separators and flippers haah_nontrivial_2023
• Theorem 1: Classification of Clifford QCAs by skew-Hermitian forms Haah2021CliffordQCA
• Lemma 1: Lemma III.8 of Ref. Haah2021CliffordQCA
• Definition 2
• Proposition 1
• proof
• Definition 3
• Lemma 2
• proof
• Definition 4