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Calibrated hypergraph states: II calibrated hypergraph state construction and applications

Roberto Zucchini

TL;DR

Calibrated hypergraph states unify higher-order quantum correlations and qudit degrees of freedom within a graded $\varOmega$-monad framework, enabling a constructive morphism-based construction of hypergraph states over general Galois rings. The paper proves that these calibrated states are locally maximally entangleable stabilizer states and clarifies their relationship to weighted hypergraph states, showing they reduce to the weighted case in qubits but not for higher-dimensional qudits. By developing the explicit calibrated hypergraph map $|{-}\rangle$ and analyzing stabilizers, monadic covariance, and effective representations, the work provides a comprehensive foundation for classification, optimization, and practical constructions (including finite-field qudits and CZ-gate assemblies). The approach yields broad generality beyond prior weighted-hypergraph formalisms, with concrete algorithms and polynomial-phase expressions when qudits are field-based, advancing both theoretical understanding and potential quantum information applications.

Abstract

Hypergraph states are a special kind of multipartite states encoded by hypergraphs relevant in quantum error correction, measurement--based quantum computation, quantum non locality and entanglement. In a series of two papers, we introduce and investigate calibrated hypergraph states, an extension of weighted hypergraph states codified by hypergraphs equipped with calibrations, a broad generalization of weightings. The guiding principle informing our approach is that a constructive theory of hypergraph states must be based on a categorical framework for both hypergraphs and multi qudit states constraining hypergraph states enough to render the determination of their general structure possible. In this second paper, we build upon the graded $\varOmega$ monadic framework worked out in the companion paper, focusing on qudits over a generic Galois ring. We explicitly construct a calibrated hypergraph state map as a special morphism of the calibrated hypergraph and multi qudit state $\varOmega$ monads. We further prove that the calibrated hypergraph states so yielded are locally maximally entangleable stabilizer states, elucidate their relationship to weighted hypergraph states, show that they reduce to the weighted ones in the familiar qubit case and prove through examples that this is no longer the case for higher qudits.

Calibrated hypergraph states: II calibrated hypergraph state construction and applications

TL;DR

Calibrated hypergraph states unify higher-order quantum correlations and qudit degrees of freedom within a graded -monad framework, enabling a constructive morphism-based construction of hypergraph states over general Galois rings. The paper proves that these calibrated states are locally maximally entangleable stabilizer states and clarifies their relationship to weighted hypergraph states, showing they reduce to the weighted case in qubits but not for higher-dimensional qudits. By developing the explicit calibrated hypergraph map and analyzing stabilizers, monadic covariance, and effective representations, the work provides a comprehensive foundation for classification, optimization, and practical constructions (including finite-field qudits and CZ-gate assemblies). The approach yields broad generality beyond prior weighted-hypergraph formalisms, with concrete algorithms and polynomial-phase expressions when qudits are field-based, advancing both theoretical understanding and potential quantum information applications.

Abstract

Hypergraph states are a special kind of multipartite states encoded by hypergraphs relevant in quantum error correction, measurement--based quantum computation, quantum non locality and entanglement. In a series of two papers, we introduce and investigate calibrated hypergraph states, an extension of weighted hypergraph states codified by hypergraphs equipped with calibrations, a broad generalization of weightings. The guiding principle informing our approach is that a constructive theory of hypergraph states must be based on a categorical framework for both hypergraphs and multi qudit states constraining hypergraph states enough to render the determination of their general structure possible. In this second paper, we build upon the graded monadic framework worked out in the companion paper, focusing on qudits over a generic Galois ring. We explicitly construct a calibrated hypergraph state map as a special morphism of the calibrated hypergraph and multi qudit state monads. We further prove that the calibrated hypergraph states so yielded are locally maximally entangleable stabilizer states, elucidate their relationship to weighted hypergraph states, show that they reduce to the weighted ones in the familiar qubit case and prove through examples that this is no longer the case for higher qudits.

Paper Structure

This paper contains 23 sections, 257 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Graph and hypergraph states
  3. Plan of the endeavour
  4. Review of the graded monadic framework of hypergraph states
  5. The calibrated hypergraph state construction
  6. Galois qudits and all that
  7. Galois rings
  8. The cyclicity monoid of a Galois ring
  9. Multi Galois cdit configuration sets as Galois ring modules
  10. Galois qudit Pauli groups
  11. Calibrated hypergraph states
  12. Calibrated hypergraph states
  13. Calibrated hypergraph states as stabilizer states
  14. Calibrated hypergraphs states and local maximal entangleability
  15. Optimizing the classification of calibrated hypergraph states
  16. ...and 8 more sections

Figures (1)

  • Figure 1: Hypergraphs can be rendered pictorially by representing vertices as dots (blue) and hyperedges as closed lines encircling the dots corresponding to their vertices (red). Panel $(a)$ shows a weighted hypergraph encoding a weighted hypergraph state. The integer juxtaposed to each hyperedge (orange) indicates its weight. Panel $(b)$ shows a calibrated hypergraph encoding a calibrated hypergraph state. The symbols attached to each vertex (teal) indicate the exponents to which the underlying qudit's Galois variable is raised. The integers collocated upon each hyperedge (orange) indicate the weights with which the corresponding exponent assignments of the encircled vertices appear. When a vertex belongs to several hyperedges, several exponent strings separated by colons are placed on it.

Theorems & Definitions (49)

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  • ...and 39 more