Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2405.09277 · Journal of High Energy Physics

Generalized cluster states from Hopf algebras: non-invertible symmetry and Hopf tensor network representation

Zhian Jia

Published Sep 20, 2024• 17 citations
arXivHTMLPDF

Abstract

A bstract Cluster states are crucial resources for measurement-based quantum computation (MBQC). It exhibits symmetry-protected topological (SPT) order, thus also playing a crucial role in studying topological phases. We present the construction of cluster states based on Hopf algebras. By generalizing the finite group valued qudit to a Hopf algebra valued qudit and introducing the generalized Pauli-X operator based on the regular action of the Hopf algebra, as well as the generalized Pauli-Z operator based on the irreducible representation action on the Hopf algebra, we develop a comprehensive theory of Hopf qudits. We demonstrate that non-invertible symmetry naturally emerges for Hopf qudits. Subsequently, for a bipartite graph termed the cluster graph, we assign the identity state and trivial representation state to even and odd vertices, respectively. Introducing the edge entangler as controlled regular action, we provide a general construction of Hopf cluster states. To ensure the commutativity of the edge entangler, we propose a method to construct a cluster lattice for any triangulable manifold. We use the 1d cluster state as an example to illustrate our construction. As this serves as a promising candidate for SPT phases, we construct the gapped Hamiltonian for this scenario and provide a detailed discussion of its non-invertible symmetries. We demonstrate that the 1d cluster state model is equivalent to the quasi-1d Hopf quantum double model with one rough boundary and one smooth boundary. We also discuss the generalization of the Hopf cluster state model to the Hopf ladder model through symmetry topological field theory. Furthermore, we introduce the Hopf tensor network representation of Hopf cluster states by integrating the tensor representation of structure constants with the string diagrams of the Hopf algebra, which can be used to solve the Hopf cluster state model.

DOI
10.1007/jhep09(2024)147

Related papers

Weak Hopf non-invertible symmetry-protected topological spin liquid and lattice realization of (1+1)D symmetry topological field theory

Zhian Jia

A bstract We propose weak Hopf symmetry as a general framework to explore (1+1)D topological phases that exhibit non-invertible symmetries. Inspired by the Symmetry Topological Field Theory (SymTFT) description of quantum phases with non-invertible symmetry, we construct a lattice model by introducing two distinct topological boundary conditions for a weak Hopf lattice gauge theory. One boundary encodes the topological symmetry information, while the other incorporates the non-topological dynamics. The resulting model is termed the cluster ladder model. We demonstrate that the cluster state model is a special case of this broader class of lattice models exhibiting weak Hopf symmetry H × Ĥ , where H is a weak Hopf algebra and Ĥ is its dual weak Hopf algebra. On a closed manifold, the symmetry reduces to Cocom( H ) × Cocom( Ĥ ), corresponding to the cocommutative subalgebras of H × Ĥ . An essential weak Hopf sub-symmetry is Cocom( H ) × Rep( H ), which, in the finite group case, reduces to the familiar symmetry G × Rep( G ). To exactly solve the lattice model, we introduce a weak Hopf tensor network. Furthermore, we demonstrate how to construct the lattice realization of an arbitrary fusion category symmetry $$ \mathcal{S} $$ S via combining Tannaka-Krein reconstruction or weak Hopf tube algebra and the cluster ladder model.

arXiv:2412.15336•Journal of High Energy Physics•Mar 16, 2026•5 citations

Generalized cluster states in 2+1d: non-invertible symmetries, interfaces, and parameterized families

Kansei Inamura, Shuhei Ohyama

We construct 2+1-dimensional lattice models of symmetry-protected topological (SPT) phases with non-invertible symmetries and investigate their properties using tensor networks. These models, which we refer to as generalized cluster models, are constructed by gauging a subgroup symmetry $H \subset G$ in models with a finite group 0-form symmetry $G$. By construction, these models have a non-invertible symmetry described by the group-theoretical fusion 2-category $\mathcal{C}(G; H)$. After identifying the tensor network representations of the symmetry operators, we study the symmetry acting on the interface between two generalized cluster states. In particular, we will see that the symmetry at the interface is described by a multifusion category known as the strip 2-algebra. By studying possible interface modes allowed by this symmetry, we show that the interface between generalized cluster states in different SPT phases must be degenerate. This result generalizes the ordinary bulk-boundary correspondence. Furthermore, we construct parameterized families of generalized cluster states and study the topological charge pumping phenomena, known as the generalized Thouless pump. We exemplify our construction with several concrete cases, and compare them with known phases, such as SPT phases with $2\mathrm{Rep}((\mathbb{Z}_{2}^{[1]}\times\mathbb{Z}_{2}^{[1]})\rtimes\mathbb{Z}_{2}^{[0]})$ symmetry.

cond-mat.str-elCross-listedarXiv:2601.08615•SciPost Physics•Jan 14, 2026•0 citations

Quantum Cluster State Spin Chain with Ising Fusion Category Symmetry: A Perspective from Weak Hopf SymTFT

Zhian Jia

In this work, we present a construction of a cluster state lattice Hamiltonian that exhibits the symmetry of the Ising fusion algebra. This construction is formulated within the framework of weak Hopf symmetry topological field theory (SymTFT), where we assign smooth and rough boundaries to the weak Hopf quantum double model, thereby extending the conventional cluster state model. Central to our construction is the weak Hopf Ising boundary tube algebra $\mathcal{T}_{\mathsf{Ising}}$, whose representation category is equivalent to the Ising fusion category $\mathsf{Ising}$. We take this algebra as the input data for the weak Hopf quantum double model. The resulting model exhibits Ising fusion symmetry on both open and closed $1\text{d}$ manifolds. On open manifolds, the symmetry is governed by $\mathcal{T}_{\mathsf{Ising}} \otimes \mathcal{T}_{\mathsf{Ising}}^{\vee}$; on closed manifolds, it reduces to $\operatorname{Cocom}(\mathcal{T}_{\mathsf{Ising}}) \otimes \operatorname{Cocom}(\mathcal{T}_{\mathsf{Ising}}^{\vee})$. Since the Ising fusion algebra embeds into $\operatorname{Cocom}(\mathcal{T}_{\mathsf{Ising}}^{\vee})$, the model faithfully realizes the symmetry of the Ising fusion category.

arXiv:2508.02424•Aug 4, 2025•0 citations

Cluster State as a Noninvertible Symmetry-Protected Topological Phase

Sahand Seifnashri, Shu-Heng Shao

We show that the standard 1+1D Z_{2}×Z_{2} cluster model has a noninvertible global symmetry, described by the fusion category Rep(D_{8}). Therefore, the cluster state is not only a Z_{2}×Z_{2} symmetry protected topological (SPT) phase, but also a noninvertible SPT phase. We further find two new commuting Pauli Hamiltonians for the other two Rep(D_{8}) SPT phases on a tensor product Hilbert space of qubits, matching the classification in field theory and mathematics. We identify the edge modes and the local projective algebras at the interfaces between these noninvertible SPT phases. Finally, we show that there does not exist a symmetric entangler that maps between these distinct SPT states.

arXiv:2404.01369•Physical Review Letters•Sep 10, 2024•69 citations

Quantum Cluster State Model with Haagerup Fusion Category Symmetry

Zhian Jia

Abstract A ‐D lattice model inspired is proposed by a weak Hopf algebra generalization of the cluster state model. This construction serves as a candidate framework for systematically realizing the Haagerup fusion category symmetry on spin chains. It is shown that the model exhibits Haagerup fusion algebra symmetry and features a tensor product Hilbert space. The construction begins with a reconstruction of the Haagerup weak Hopf algebra from the Haagerup fusion category , ensuring that the representation category of is equivalent to . Utilizing the framework of symmetry topological field theory (SymTFT), an ultra‐thin weak Hopf quantum double model that has a smooth topological boundary condition is developed. It is shown that this model supports Haagerup fusion algebra symmetry. Finally, the ground state of the model is solved for in terms of a weak Hopf matrix product state, which serves as a natural generalization of the cluster state, embodying Haagerup fusion algebra symmetry. Since the Haagerup fusion category is an anomalous symmetry, the model presented realizes the gapped Haagerup spontaneous symmetry breaking (SSB) phase.

arXiv:2412.19657•Advanced Quantum Technologies•Oct 30, 2025•8 citations

Non-invertible SPTs: an on-site realization of (1+1)d anomaly-free fusion category symmetry

Meng, Chenqi, Yang, Xinping, Tian Lan +1 more

We investigate (1+1)d symmetry-protected topological (SPT) phases with fusion category symmetries. We emphasize that the UV description of an anomaly-free fusion category symmetry must include the fiber functor, giving rise to a local symmetry action, a charge category and a trivial phase. We construct an ``onsite'' matrix-product-operator (MPO) version of the Hopf algebra symmetry operators in a lattice model with tensor-product Hilbert space. In particular, we propose a systematic framework for classifying and constructing SPTs with non-invertible symmetries. An SPT phase corresponds to a Q-system in the charge category, such that the Q-system becomes a matrix algebra when the symmetry is forgotten. As an example, we provide an explicit microscopic realization of all three $\mathsf{Rep}^\dagger(D_8)$ SPT phases, including a trivial phase, and further demonstrate the $S_3$-duality among these three SPT phases.

arXiv:2412.20546•Dec 29, 2024•5 citations
A bstract Cluster states are crucial resources for measurement-based quantum computation (MBQC). It exhibits symmetry-protected topological (SPT) order, thus also playing a crucial role in studying topological phases. We present the construction of cluster states based on Hopf algebras. By generalizing the finite group valued qudit to a Hopf algebra valued qudit and introducing the generalized Pauli-X operator based on the regular action of the Hopf algebra, as well as the generalized Pauli-Z operator based on the irreducible representation action on the Hopf algebra, we develop a comprehensive theory of Hopf qudits. We demonstrate that non-invertible symmetry naturally emerges for Hopf qudits. Subsequently, for a bipartite graph termed the cluster graph, we assign the identity state and trivial representation state to even and odd vertices, respectively. Introducing the edge entangler as controlled regular action, we provide a general construction of Hopf cluster states. To ensure the commutativity of the edge entangler, we propose a method to construct a cluster lattice for any triangulable manifold. We use the 1d cluster state as an example to illustrate our construction. As this serves as a promising candidate for SPT phases, we construct the gapped Hamiltonian for this scenario and provide a detailed discussion of its non-invertible symmetries. We demonstrate that the 1d cluster state model is equivalent to the quasi-1d Hopf quantum double model with one rough boundary and one smooth boundary. We also discuss the generalization of the Hopf cluster state model to the Hopf ladder model through symmetry topological field theory. Furthermore, we introduce the Hopf tensor network representation of Hopf cluster states by integrating the tensor representation of structure constants with the string diagrams of the Hopf algebra, which can be used to solve the Hopf cluster state model.
A bstract We propose weak Hopf symmetry as a general framework to explore (1+1)D topological phases that exhibit non-invertible symmetries. Inspired by the Symmetry Topological Field Theory (SymTFT) description of quantum phases with non-invertible symmetry, we construct a lattice model by introducing two distinct topological boundary conditions for a weak Hopf lattice gauge theory. One boundary encodes the topological symmetry information, while the other incorporates the non-topological dynamics. The resulting model is termed the cluster ladder model. We demonstrate that the cluster state model is a special case of this broader class of lattice models exhibiting weak Hopf symmetry H × Ĥ , where H is a weak Hopf algebra and Ĥ is its dual weak Hopf algebra. On a closed manifold, the symmetry reduces to Cocom( H ) × Cocom( Ĥ ), corresponding to the cocommutative subalgebras of H × Ĥ . An essential weak Hopf sub-symmetry is Cocom( H ) × Rep( H ), which, in the finite group case, reduces to the familiar symmetry G × Rep( G ). To exactly solve the lattice model, we introduce a weak Hopf tensor network. Furthermore, we demonstrate how to construct the lattice realization of an arbitrary fusion category symmetry S \mathcal{S} S S via combining Tannaka-Krein reconstruction or weak Hopf tube algebra and the cluster ladder model.
We construct 2+1-dimensional lattice models of symmetry-protected topological (SPT) phases with non-invertible symmetries and investigate their properties using tensor networks. These models, which we refer to as generalized cluster models, are constructed by gauging a subgroup symmetry H⊂GH \subset GH⊂G in models with a finite group 0-form symmetry GGG. By construction, these models have a non-invertible symmetry described by the group-theoretical fusion 2-category C(G;H)\mathcal{C}(G; H)C(G;H). After identifying the tensor network representations of the symmetry operators, we study the symmetry acting on the interface between two generalized cluster states. In particular, we will see that the symmetry at the interface is described by a multifusion category known as the strip 2-algebra. By studying possible interface modes allowed by this symmetry, we show that the interface between generalized cluster states in different SPT phases must be degenerate. This result generalizes the ordinary bulk-boundary correspondence. Furthermore, we construct parameterized families of generalized cluster states and study the topological charge pumping phenomena, known as the generalized Thouless pump. We exemplify our construction with several concrete cases, and compare them with known phases, such as SPT phases with 2Rep((Z2[1]×Z2[1])⋊Z2[0])2\mathrm{Rep}((\mathbb{Z}_{2}^{[1]}\times\mathbb{Z}_{2}^{[1]})\rtimes\mathbb{Z}_{2}^{[0]})2Rep((Z2[1]​×Z2[1]​)⋊Z2[0]​) symmetry.
In this work, we present a construction of a cluster state lattice Hamiltonian that exhibits the symmetry of the Ising fusion algebra. This construction is formulated within the framework of weak Hopf symmetry topological field theory (SymTFT), where we assign smooth and rough boundaries to the weak Hopf quantum double model, thereby extending the conventional cluster state model. Central to our construction is the weak Hopf Ising boundary tube algebra TIsing\mathcal{T}_{\mathsf{Ising}}TIsing​, whose representation category is equivalent to the Ising fusion category Ising\mathsf{Ising}Ising. We take this algebra as the input data for the weak Hopf quantum double model. The resulting model exhibits Ising fusion symmetry on both open and closed 1d1\text{d}1d manifolds. On open manifolds, the symmetry is governed by TIsing⊗TIsing∨\mathcal{T}_{\mathsf{Ising}} \otimes \mathcal{T}_{\mathsf{Ising}}^{\vee}TIsing​⊗TIsing∨​; on closed manifolds, it reduces to Cocom⁡(TIsing)⊗Cocom⁡(TIsing∨)\operatorname{Cocom}(\mathcal{T}_{\mathsf{Ising}}) \otimes \operatorname{Cocom}(\mathcal{T}_{\mathsf{Ising}}^{\vee})Cocom(TIsing​)⊗Cocom(TIsing∨​). Since the Ising fusion algebra embeds into Cocom⁡(TIsing∨)\operatorname{Cocom}(\mathcal{T}_{\mathsf{Ising}}^{\vee})Cocom(TIsing∨​), the model faithfully realizes the symmetry of the Ising fusion category.
We show that the standard 1+1D Z_{2}×Z_{2} cluster model has a noninvertible global symmetry, described by the fusion category Rep(D_{8}). Therefore, the cluster state is not only a Z_{2}×Z_{2} symmetry protected topological (SPT) phase, but also a noninvertible SPT phase. We further find two new commuting Pauli Hamiltonians for the other two Rep(D_{8}) SPT phases on a tensor product Hilbert space of qubits, matching the classification in field theory and mathematics. We identify the edge modes and the local projective algebras at the interfaces between these noninvertible SPT phases. Finally, we show that there does not exist a symmetric entangler that maps between these distinct SPT states.
Abstract A ‐D lattice model inspired is proposed by a weak Hopf algebra generalization of the cluster state model. This construction serves as a candidate framework for systematically realizing the Haagerup fusion category symmetry on spin chains. It is shown that the model exhibits Haagerup fusion algebra symmetry and features a tensor product Hilbert space. The construction begins with a reconstruction of the Haagerup weak Hopf algebra from the Haagerup fusion category , ensuring that the representation category of is equivalent to . Utilizing the framework of symmetry topological field theory (SymTFT), an ultra‐thin weak Hopf quantum double model that has a smooth topological boundary condition is developed. It is shown that this model supports Haagerup fusion algebra symmetry. Finally, the ground state of the model is solved for in terms of a weak Hopf matrix product state, which serves as a natural generalization of the cluster state, embodying Haagerup fusion algebra symmetry. Since the Haagerup fusion category is an anomalous symmetry, the model presented realizes the gapped Haagerup spontaneous symmetry breaking (SSB) phase.
We investigate (1+1)d symmetry-protected topological (SPT) phases with fusion category symmetries. We emphasize that the UV description of an anomaly-free fusion category symmetry must include the fiber functor, giving rise to a local symmetry action, a charge category and a trivial phase. We construct an ``onsite'' matrix-product-operator (MPO) version of the Hopf algebra symmetry operators in a lattice model with tensor-product Hilbert space. In particular, we propose a systematic framework for classifying and constructing SPTs with non-invertible symmetries. An SPT phase corresponds to a Q-system in the charge category, such that the Q-system becomes a matrix algebra when the symmetry is forgotten. As an example, we provide an explicit microscopic realization of all three Rep†(D8)\mathsf{Rep}^\dagger(D_8)Rep†(D8​) SPT phases, including a trivial phase, and further demonstrate the S3S_3S3​-duality among these three SPT phases.