We prove that the three-dimensional chiral $\Z_p$ 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.
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The paper asks whether known nontrivial three-dimensional Clifford quantum cellular automata (QCA) remain nontrivial when arbitrary, non-Clifford finite-depth circuits and ancillary systems are allowed. It proves that the chiral $\mathbb Z_p$ constructions are not stabilized circuits, even modulo translations or generalized shifts, for every odd prime $p$ and every parameter $k\in\mathbb F_p^\times$. Previously, their nontriviality was established only within the Clifford category.
The argument reduces circuit nontriviality to nonvanishing of a two-dimensional invertible subalgebra’s Brauer class. It introduces a square-class invariant $I(\mathcal B)\in\mathbb Q_{>0}/\mathbb Q_{>0}^2$: exchange two copies of $\mathcal B$ while fixing their commutants, truncate this involution to a half-plane, and take the GNVW index of the truncation squared. The resulting class is independent of the truncation, additive, and vanishes on stably trivial subalgebras. An explicit Clifford truncation gives $\operatorname{ind}(t^2)=p^{-1}$, hence $I(\mathcal B_{p,k})=[p]\ne0$. A general computation relates the invariant to the parity of a half-plane compression’s cokernel dimension. Consequently, the three-dimensional QCA quotient is not finitely generated; the boundary Brauer group contains $\bigoplus_{p\equiv1\pmod4}\mathbb Z/2$.
The construction requires uniformly controlled truncations, supplied explicitly for these examples; general existence after stabilization invokes the classification of two-dimensional QCA. The invariant detects only square-class information and misses even-rank Clifford classes. It is not expected to detect the three-fermion QCA. Its proposed identification with squared total quantum dimension remains conjectural beyond the established cases.
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The computations in Section~\ref{sec:examples} were carried out with the assistance of Claude Fable~5.1 and Claude Opus~5.5 (Anthropic), which were also used in preparing the text.
Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.
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.
The paper asks whether known nontrivial three-dimensional Clifford quantum cellular automata (QCA) remain nontrivial when arbitrary, non-Clifford finite-depth circuits and ancillary systems are allowed. It proves that the chiral Zp constructions are not stabilized circuits, even modulo translations or generalized shifts, for every odd prime p and every parameter k∈Fp×. Previously, their nontriviality was established only within the Clifford category.
The argument reduces circuit nontriviality to nonvanishing of a two-dimensional invertible subalgebra’s Brauer class. It introduces a square-class invariant I(B)∈Q>0/Q>02: exchange two copies of B while fixing their commutants, truncate this involution to a half-plane, and take the GNVW index of the truncation squared. The resulting class is independent of the truncation, additive, and vanishes on stably trivial subalgebras. An explicit Clifford truncation gives ind(t2)=p−1, hence I(Bp,k)=[p]=0. A general computation relates the invariant to the parity of a half-plane compression’s cokernel dimension. Consequently, the three-dimensional QCA quotient is not finitely generated; the boundary Brauer group contains ⨁p≡1(mod4)Z/2.
The construction requires uniformly controlled truncations, supplied explicitly for these examples; general existence after stabilization invokes the classification of two-dimensional QCA. The invariant detects only square-class information and misses even-rank Clifford classes. It is not expected to detect the three-fermion QCA. Its proposed identification with squared total quantum dimension remains conjectural beyond the established cases.
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.
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.
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.
Self-dualities and the stacking of symmetry-protected topological (SPT) phases are basic operations on quantum many-body systems. For a Zp one-form symmetry in 3+1d these correspond to the Kramers-Wannier-Wegner duality S, which is the gauging operation underlying non-invertible duality symmetries, and the stacking of a 1-form symmetry SPT T. In the continuum, they form a central extension of PSL(2,Z4) for p=2, and of SL(2,Zp) for odd primes p, whose central elements are invertible theories with purely gravitational response. These central extensions are governed by a twisted, graded generalization of the Witt group of abelian anyon theories, which we determine. For p=2 the resulting group is the single-qubit Clifford group, with duality and entangler acting as the Hadamard and phase gates. We realize this entire structure microscopically as quantum cellular automata (QCA) acting on a certain local operator algebra associated with a spin lattice Hilbert space on a cubic lattice. Specifically, our local operator algebra is built by starting with all local operators commuting with a Zp 1-form symmetry, and taking the quotient by all the (local) 1-form symmetry generators. The central elements can always be extended to the full tensor product algebra with a uniquely defined QCA class. For p=2 they are generated by the non-trivial semion QCA, and for odd prime p they are generated by the non-trivial Zp Clifford QCA. Consequently the lattice fusion rules reproduce the continuum ones only up to these QCAs and lattice translations, giving rise to fusion rules refined by QCAs.
We construct a novel three-dimensional quantum cellular automaton (QCA) based on a system with short-range entangled bulk and chiral semion boundary topological order.We argue that either the QCA is nontrivial, i.e., not a finite-depth circuit of local quantum gates, or there exists a two-dimensional commuting projector Hamiltonian realizing the chiral semion topological order [characterized by U(1) 2 Chern-Simons theory].Our QCA is obtained by first constructing the Walker-Wang Hamiltonian of a certain premodular tensor category of order four, then condensing the deconfined bulk boson at the level of lattice operators.We show that the resulting Hamiltonian hosts chiral semion surface topological order in the presence of a boundary and can be realized as a non-Pauli stabilizer code on qubits, from which the QCA is defined.The construction is then generalized to a class of QCAs defined by non-Pauli stabilizer codes on 2 n -dimensional qudits that feature surface anyons described by U(1) 2 n Chern-Simons theory.Our results support the conjecture that the group of nontrivial three-dimensional QCAs is isomorphic to the Witt group of nondegenerate braided fusion categories.
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.