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

Abstract

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.

v1: 41 pages; v2: 46 pages, several typos are corrected, a more detailed discussion of non-invertible symmetry is provided; v3: typos are corrected, more discussion on Hopf symmetries added

References in corpus (23)

• Generalized Global Symmetries
• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• Interacting anyons in topological quantum liquids: The golden chain
• Generalized Symmetries in Condensed Matter
• Symmetries and Strings of Adjoint QCD${}_2$
• Orbifold groupoids
• Quantum secret sharing with qudit graph states
• Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry
• Topological defects for the free boson CFT
• Quantum Error Correcting Codes Using Qudit Graph States
• Lecture Notes on Generalized Symmetries and Applications
• On lattice models of gapped phases with fusion category symmetries
• Cluster state as a non-invertible symmetry protected topological phase
• Higher categorical symmetries and gauging in two-dimensional spin systems
• Classifying phases protected by matrix product operator symmetries using matrix product states
• Fermionization of fusion category symmetries in 1+1 dimensions
• Generalized Cluster States Based on Finite Groups
• Ribbon operators in the generalized Kitaev quantum double model based on Hopf algebras
• Boundary and domain wall theories of 2d generalized quantum double model
• On weak Hopf symmetry and weak Hopf quantum double model
• Invertible bimodule categories and generalized Schur orthogonality
• Weak Hopf symmetry and tube algebra of the generalized multifusion string-net model
• Electric-magnetic duality and ${\mathbb{Z}}_2$ symmetry enriched Abelian lattice gauge theory

Cited by in corpus (14)

• Non-invertible and higher-form symmetries in 2+1d lattice gauge theories
• (SPT-)LSM theorems from projective non-invertible symmetries
• Gauging non-invertible symmetries on the lattice
• Global symmetries of quantum lattice models under non-invertible dualities
• Quantum Cluster State Model with Haagerup Fusion Category Symmetry
• Duality-preserving deformation of 3+1d lattice ${\mathbb{Z}}_2$ gauge theory with exact gapped ground states
• Noninvertible symmetry and topological holography for modulated SPT in one dimension
• Weak Hopf non-invertible symmetry-protected topological spin liquid and lattice realization of (1+1)D symmetry topological field theory
• (2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry
• Weak Hopf tube algebra for domain walls between 2d gapped phases of Turaev-Viro TQFTs
• Subsystem Symmetry-Protected Topological Phases from Subsystem SymTFT of 2-Foliated Exotic Tensor Gauge Theory
• Antilinear superoperator, quantum geometric invariance, and antilinear symmetry for higher-dimensional quantum systems
• Higher-order topological phases protected by noninvertible and subsystem symmetries
• Spontaneously Broken Non-Invertible Symmetries in Transverse-Field Ising Qudit Chains