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.
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.
Quantum cellular automata (QCAs) describe locality-preserving quantum dynamics and link quantum information, many-body physics, and topological quantum field theory (TQFT). Constructing a QCA from a TQFT, however, remains challenging: although a topological action yields a Hamiltonian with commuting terms that realizes the desired ground state, it does not by itself specify an automorphism of the full operator algebra. We develop a unified algebraic construction that extends such Hamiltonians to complete separator-flipper algebras on the full tensor-product Hilbert space, thereby defining the corresponding QCAs microscopically. In 3 spatial dimensions, our framework unifies all previously known QCA constructions associated with the Z8×Z2 subgroup of the Witt group, including the U(1)2 and U(1)4 QCAs. The same algebraic structure yields new infinite families of generalized U(1)2 and U(1)4 non-Clifford QCAs in spatial dimensions d=4k−1. We also reformulate the 4-dimensional w2w3 QCA and use it to construct QCAs from TQFTs associated with arbitrary products of Stiefel--Whitney classes. For example, we explicitly construct two infinite families: w2nw3m QCAs in dimension d=2n+3m−1 and w2w4k−1 QCAs in dimension d=4k. For the 5-dimensional w32 and w23 QCAs, our construction gives finite-depth quantum circuits, consistent with the cobordism classification. More generally, we show that our construction yields a finite-depth quantum circuit whenever the corresponding Stiefel-Whitney number vanishes on every closed oriented manifold. Overall, these results convert invertible TQFTs into microscopic QCAs, provide higher-dimensional constructions beyond the Clifford setting, and open a systematic route to characterizing their stable structures and boundary anomalies.
Chiral topological order is incompatible with strictly two-dimensional local commuting-projector Hamiltonians on finite-dimensional tensor-product Hilbert spaces. Invertible subalgebras (ISAs) provide an alternative framework. We construct an ISA supporting chiral semions, together with its anti-semion commutant, on a square lattice with one qubit per edge. We give explicit non-Pauli generators for both algebras and prove that together they generate all local operators on the Hilbert space. Starting from the Z4 toric code, we condense the boson e2m2 and identify the semion and anti-semion algebras. We then decompose each Z4 qudit into two qubits and use a finite-depth circuit to disentangle one qubit per edge fixed by the condensation constraints. Pumping the semion ISA along the z direction defines an explicit three-dimensional non-Clifford quantum cellular automaton (QCA). We show that pumping eight times is equivalent to a translation followed by a finite-depth quantum circuit with ancillary qubits. We also give an explicit finite-depth circuit relating this construction to the semion QCA previously obtained from topological quantum field theory. A similar condensation construction yields a U(1)4 ISA on a square lattice. Together with the semion and earlier odd-prime ISAs, finite stacking realizes every pointed Witt class of bosonic Abelian anyon theories. Finally, we extend both constructions to all spatial dimensions d=4k−2, with one qubit or Z4 qudit on each (2k−1)-cell for the semion or U(1)4 ISA, respectively.
Recent work has constructed higher-dimensional analogs of non-invertible symmetries similar to 1+1d Kramers-Wannier duality. Although their continuum descriptions often treat purely gravitational topological terms as inessential counterterms, these terms can have an essential lattice manifestation: they distinguish states prepared by finite-depth quantum circuits (FDQCs) from those entangled by nontrivial quantum cellular automata (QCAs). Motivated by this mismatch, we show that QCAs associated with gravitational topological responses arise in several related settings: (1) lattice realizations of projective SL(2,ZN) relations generated by topological operations on symmetries; (2) squares of dualities that generalize the relation between fermionization and Kramers-Wannier duality; (3) lattice implementations of QCAs through higher-form gauging; and (4) invertible phases protected by generalized time-reversal symmetries. We derive new projective SL(2,ZN) relations whose projective phases are gravitational topological responses constructed from Stiefel-Whitney classes. We furthermore give a general protocol for preparing the associated QCA-entangled states using finite-depth unitary circuits, measurements, and error correction. These results unify the study of gravitational topological responses in field theories, higher dimensional dualities, and quantum cellular automata.
We prove that the three-dimensional chiral Zp Clifford quantum cellular automata of Haah and of Sun--Yang--Wang--Tantivasadakarn--Chen are, for every odd prime p, not finite-depth quantum circuits, even after tensoring with arbitrary ancillary degrees of freedom and composing with translations. Previously such statements were known only inside the Clifford category. As a consequence, the group of three-dimensional QCA modulo stabilized circuits and translations is not finitely generated. The key ingredient is a new invariant of invertible subalgebras, which we conjecture to be the square of the total quantum dimension of the hosted anyon theory, modulo rational squares.
We study locality preserving automorphisms of operator algebras on D-dimensional uniform lattices of prime p-dimensional qudits quantum cellular automata (QCAs), specializing in those that are translation invariant (TI), and map every prime p-dimensional Pauli matrix to a tensor product of Pauli matrices (Clifford). We associate antihermitian forms of the unit determinant over Laurent polynomial rings to TI Clifford QCA with lattice boundaries and prove that the form determines the QCA up to Clifford circuits and shifts (trivial). It follows that every 2D TI Clifford QCA is trivial since the antihermitian form in this case is always trivial. Furthermore, we prove that for any D, the fourth power of any TI Clifford QCA is trivial. We present explicit examples of nontrivial TI Clifford QCA for D = 3 and any odd prime p and show that the Witt group of the finite field Fp is a subgroup of the group C(D=3,p) of all TI Clifford QCA modulo trivial ones. That is, C(D=3,p≡1mod4)⊇Z2×Z2 and C(D=3,p≡3mod4)⊇Z4. The examples are found by disentangling the ground state of a commuting Pauli Hamiltonian, which is constructed by coupling layers of prime dimensional toric codes such that an exposed surface has an anomalous topological order that is not realizable by commuting Pauli Hamiltonians strictly in two dimensions. In an appendix independent of the main body of this paper, we revisit a recent theorem of Freedman and Hastings that any two-dimensional QCA, which is not necessarily Clifford or translation invariant, is a constant depth quantum circuit followed by a shift. We give a more direct proof of the theorem without using any ancillas.
Over an arbitrary commutative ring R, we develop a theory of quantum cellular automata. We then use algebraic K-theory to construct a space Q(X) of quantum cellular automata (QCA) on a given metric space X. In most cases of interest, π0Q(X) classifies QCA up to quantum circuits and stabilization. Notably, the QCA spaces are related by homotopy equivalences Q(∗)≃ΩnQ(Zn) for all n, which shows that the classification of QCA on Euclidean lattices is given by an Ω-spectrum indexed by the dimension n. As a corollary, we also obtain a non-connective delooping of the K-theory of Azumaya R-algebras, which may be of independent interest. We also include a section leading to the Ω-spectrum for QCA over C∗-algebras with unitary circuits.