  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 delve
into a detailed discussion of its non-invertible symmetries. We also show that
the 1d cluster state model is equivalent to the quasi-1d Hopf quantum double
model. We also 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.
