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arXiv:2609.09317

Fermionic quantum cellular automata in 2d are trivial

Jeffrey Kwan, David M. Long, Jeongwan Haah

math-phcond-mat.str-elmath.OAquant-phSubmitted Sep 8, 2026• Announced Sep 10, 2026• 0 citations
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Abstract

Fermionic quantum cellular automata (QCA) are automorphisms of local fermionic operator algebras ($\mathbb Z_2$-graded superalgebras) that have bounded spread: they map local operators to nearby operators. We prove that every 2-dimensional fermionic QCA on a locally finite-dimensional algebra is a composition of local automorphisms and a fermionic shift. This is implied by our result that every locally finite-dimensional fermionic invertible subalgebra in a one-dimensional lattice is Brauer trivial, \textit{i.e.}, it is stably bounded-spread isomorphic to a tensor product fermionic algebra.

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Quantum cellular automata and invertible phases of matter

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We introduce and study (fermionic and bosonic) invertible quasi-local algebras over uniformly locally finite metric spaces $X$ with infinite-dimensional local von Neumann algebras. We show that the group of Brauer equivalence classes of such algebras is isomorphic to both the group of phases of invertible states and the group of stable equivalence classes of quantum cellular automata over $X\times \mathbb{Z}$. Using K-theory of the symmetric monoidal category of invertible quasi-local algebras and bounded spread isomorphisms, we propose a definition of an $Ω$-spectrum of invertible phases as conjectured by Kitaev. We then show that the $c=\frac{1}{2}$ chiral Majorana fermion net and the $(E_{8})_{1}$ conformal net provide Brauer non-trivial invertible quasi-local algebras, thus providing explicit constructions of non-trivial invertible states and quantum cellular automata on $\mathbb{Z}^{2}$. In addition, we show that the time-slice nets of rational diagonal conformal field theories admit lattice degrees of freedom, which implies the discretization of any holomorphic conformal net is invertible.

math-phCross-listedarXiv:2608.26456•Aug 28, 2026•0 citations

Approximate QCAs in one dimension using approximate algebras

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Quantum cellular automata (QCAs) are automorphisms of tensor product algebras that preserve locality, with local quantum circuits as a simple example. We study approximate QCAs, where the locality condition is only satisfied up to a small error, as occurs for local quantum dynamics on the lattice. A priori, approximate QCAs could exhibit genuinely new behavior, failing to be well-approximated by any exact QCA. We show this does not occur in one dimension: every approximate QCA on a finite circle can be rounded to a strict QCA with approximately the same action on local operators, so these systems are classified by the same index as in the exact case. Previous work considered the case of the infinite line, by using global methods not amenable to finite systems. Our new approach proceeds locally and now applies to finite systems, including circles or homomorphisms from sub-intervals. We extract exact local boundary algebras from the approximate QCA restricted to local patches, then glue these to form a strict QCA. The key technical ingredient is a robust notion of the intersection of two subalgebras: when the projections onto two subalgebras approximately commute, we construct an exact subalgebra that serves as a stable proxy for their intersection. This construction uses a recent theorem of Kitaev on the rigidity of approximate $C^*$-algebras.

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Clifford quantum cellular automata: Trivial group in 2D and Witt group in 3D

Jeongwan Haah

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.

arXiv:1907.02075•Journal of Mathematical Physics•Sep 1, 2021•48 citations

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

Meng Sun, Zongyuan Wang, Bowen Yang +2 more

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 $\mathbb{Z}_8\times\mathbb{Z}_2$ subgroup of the Witt group, including the $U(1)_2$ and $U(1)_4$ QCAs. The same algebraic structure 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 $w_2w_3$ 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: $w_2^nw_3^m$ QCAs in dimension $d=2n+3m-1$ and $w_2w_{4k-1}$ QCAs in dimension $d=4k$. For the 5-dimensional $w_3^2$ and $w_2^3$ 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.

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We investigate quantum cellular automata (QCA) on one-dimensional spin systems defined over a subalgebra of the full local operator algebra – the symmetric subalgebra under a finite Abelian group symmetry G . For systems where each site carries a regular representation of G , we establish a complete classification of such subalgebra QCAs based on two topological invariants: (1) a surjective homomorphism from the group of subalgebra QCAs to the group of anyon permutation symmetries in a ( 2 + 1 ) d G gauge theory; and (2) a generalization of the Gross-Nesme-Vogts-Werner (GNVW) index that characterizes the flow of the symmetric subalgebra. Specifically, two subalgebra QCAs correspond to the same anyon permutation and share the same index if and only if they differ by a finite-depth unitary circuit composed of G -symmetric local gates. We also identify a set of operations that generate all subalgebra QCAs through finite compositions. As an example, we examine the Kramers-Wannier duality on a Z 2 symmetric subalgebra, demonstrating that it maps to the e - m permutation in the two-dimensional toric code and has an irrational index of 2 . Therefore, it cannot be extended to a QCA over the full local operator algebra and mixes nontrivially with lattice translations.

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Non-Invertible Symmetries Mixing with Witt Non-Trivial Quantum Cellular Automata

Kansei Inamura, Oskar Wojdel, Lukasz Fidkowski +1 more

Self-dualities and the stacking of symmetry-protected topological (SPT) phases are basic operations on quantum many-body systems. For a $\mathbb{Z}_p$ 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 $PSL(2,\mathbb{Z}_4)$ for $p=2$, and of $SL(2,\mathbb{Z}_p)$ 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 $\mathbb{Z}_p$ 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 $\mathbb{Z}_p$ 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.

quant-pharXiv:2607.21698•Jul 27, 2026•0 citations
Fermionic quantum cellular automata (QCA) are automorphisms of local fermionic operator algebras (Z2\mathbb Z_2Z2​-graded superalgebras) that have bounded spread: they map local operators to nearby operators. We prove that every 2-dimensional fermionic QCA on a locally finite-dimensional algebra is a composition of local automorphisms and a fermionic shift. This is implied by our result that every locally finite-dimensional fermionic invertible subalgebra in a one-dimensional lattice is Brauer trivial, \textit{i.e.}, it is stably bounded-spread isomorphic to a tensor product fermionic algebra.
We introduce and study (fermionic and bosonic) invertible quasi-local algebras over uniformly locally finite metric spaces XXX with infinite-dimensional local von Neumann algebras. We show that the group of Brauer equivalence classes of such algebras is isomorphic to both the group of phases of invertible states and the group of stable equivalence classes of quantum cellular automata over X×ZX\times \mathbb{Z}X×Z. Using K-theory of the symmetric monoidal category of invertible quasi-local algebras and bounded spread isomorphisms, we propose a definition of an ΩΩΩ-spectrum of invertible phases as conjectured by Kitaev. We then show that the c=12c=\frac{1}{2}c=21​ chiral Majorana fermion net and the (E8)1(E_{8})_{1}(E8​)1​ conformal net provide Brauer non-trivial invertible quasi-local algebras, thus providing explicit constructions of non-trivial invertible states and quantum cellular automata on Z2\mathbb{Z}^{2}Z2. In addition, we show that the time-slice nets of rational diagonal conformal field theories admit lattice degrees of freedom, which implies the discretization of any holomorphic conformal net is invertible.
Quantum cellular automata (QCAs) are automorphisms of tensor product algebras that preserve locality, with local quantum circuits as a simple example. We study approximate QCAs, where the locality condition is only satisfied up to a small error, as occurs for local quantum dynamics on the lattice. A priori, approximate QCAs could exhibit genuinely new behavior, failing to be well-approximated by any exact QCA. We show this does not occur in one dimension: every approximate QCA on a finite circle can be rounded to a strict QCA with approximately the same action on local operators, so these systems are classified by the same index as in the exact case. Previous work considered the case of the infinite line, by using global methods not amenable to finite systems. Our new approach proceeds locally and now applies to finite systems, including circles or homomorphisms from sub-intervals. We extract exact local boundary algebras from the approximate QCA restricted to local patches, then glue these to form a strict QCA. The key technical ingredient is a robust notion of the intersection of two subalgebras: when the projections onto two subalgebras approximately commute, we construct an exact subalgebra that serves as a stable proxy for their intersection. This construction uses a recent theorem of Kitaev on the rigidity of approximate C∗C^*C∗-algebras.
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\mathbb{Z}_8\times\mathbb{Z}_2Z8​×Z2​ subgroup of the Witt group, including the U(1)2U(1)_2U(1)2​ and U(1)4U(1)_4U(1)4​ QCAs. The same algebraic structure yields new infinite families of generalized U(1)2U(1)_2U(1)2​ and U(1)4U(1)_4U(1)4​ non-Clifford QCAs in spatial dimensions d=4k−1d=4k-1d=4k−1. We also reformulate the 4-dimensional w2w3w_2w_3w2​w3​ 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: w2nw3mw_2^nw_3^mw2n​w3m​ QCAs in dimension d=2n+3m−1d=2n+3m-1d=2n+3m−1 and w2w4k−1w_2w_{4k-1}w2​w4k−1​ QCAs in dimension d=4kd=4kd=4k. For the 5-dimensional w32w_3^2w32​ and w23w_2^3w23​ 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.
We investigate quantum cellular automata (QCA) on one-dimensional spin systems defined over a subalgebra of the full local operator algebra – the symmetric subalgebra under a finite Abelian group symmetry G . For systems where each site carries a regular representation of G , we establish a complete classification of such subalgebra QCAs based on two topological invariants: (1) a surjective homomorphism from the group of subalgebra QCAs to the group of anyon permutation symmetries in a ( 2 + 1 ) d G gauge theory; and (2) a generalization of the Gross-Nesme-Vogts-Werner (GNVW) index that characterizes the flow of the symmetric subalgebra. Specifically, two subalgebra QCAs correspond to the same anyon permutation and share the same index if and only if they differ by a finite-depth unitary circuit composed of G -symmetric local gates. We also identify a set of operations that generate all subalgebra QCAs through finite compositions. As an example, we examine the Kramers-Wannier duality on a Z 2 symmetric subalgebra, demonstrating that it maps to the e - m permutation in the two-dimensional toric code and has an irrational index of 2 . Therefore, it cannot be extended to a QCA over the full local operator algebra and mixes nontrivially with lattice translations.
Self-dualities and the stacking of symmetry-protected topological (SPT) phases are basic operations on quantum many-body systems. For a Zp\mathbb{Z}_pZp​ one-form symmetry in 3+1d these correspond to the Kramers-Wannier-Wegner duality SSS, which is the gauging operation underlying non-invertible duality symmetries, and the stacking of a 1-form symmetry SPT TTT. In the continuum, they form a central extension of PSL(2,Z4)PSL(2,\mathbb{Z}_4)PSL(2,Z4​) for p=2p=2p=2, and of SL(2,Zp)SL(2,\mathbb{Z}_p)SL(2,Zp​) for odd primes ppp, 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=2p=2p=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\mathbb{Z}_pZp​ 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=2p=2p=2 they are generated by the non-trivial semion QCA, and for odd prime ppp they are generated by the non-trivial Zp\mathbb{Z}_pZp​ 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.