Quantum hypergraph states in continuous variables

Moore, Darren W.

doi:10.1103/physreva.100.062301

Cited by 5 publications

(2 citation statements)

References 57 publications

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“…Graph states [5] are very important in quantum computation and quantum errorcorrecting, and are the base of measurement-based quantum computation. There are several generalizations of graph states to various types of graphs, like hypergraphs (a graph in which an edge can join any number of vertices) and we refer here to [3,7,9] and the references therein. Certainly, a natural question that arises would be how to extend the concept of graph states to quantum states associated with mixed graphs, which are graphs where some or all of the edges may be directed (and no multiple edges or loops are allowed).…”

Section: Introductionmentioning

confidence: 99%

Quantum states associated to mixed graphs and their algebraic characterization

Riera¹,

Parker²,

Stănică³

2023

AMC

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Graph states are present in quantum information and found applications ranging from quantum network protocols (like secret sharing) to measurement based quantum computing. In this paper, we extend the notion of graph states, which can be regarded as pure quantum graph states, or as homogeneous quadratic Boolean functions associated to simple undirected graphs, to quantum states based on mixed graphs (graphs which allow both directed and undirected edges), obtaining mixed quantum states, which are defined by matrices associated to the measurement of homogeneous quadratic Boolean functions in some (ancillary) variables. In our main result, we describe the extended graph state as the sum of terms of a commutative subgroup of the stabilizer group of the corresponding mixed graph with the edges' directions reversed.

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Section: Introductionmentioning

confidence: 99%

Quantum states associated to mixed graphs and their algebraic characterization

Riera¹,

Parker²,

Stănică³

2023

AMC

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“…Otherwise, the verifier rejects. This scheme also works for verification of non-Gaussian CV hypergraph states [10,23]. When the target state is a hypergraph state, the verifier follows the same procedure, except that qj := U qj U † and pj := U pj U † are different from above.…”

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confidence: 99%

Efficient verification of continuous-variable quantum states and devices without assuming identical and independent operations

Wu,

Bai,

Chiribella

et al. 2020

Preprint

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

Jia

2024

J. High Energ. Phys.

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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.

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Quantum hypergraph states in continuous variables

Cited by 5 publications

References 57 publications

Quantum states associated to mixed graphs and their algebraic characterization

Quantum states associated to mixed graphs and their algebraic characterization

Efficient verification of continuous-variable quantum states and devices without assuming identical and independent operations

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

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