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Vertex-Minor Universal Graphs for Generating Entangled Quantum Subsystems

Authors Maxime Cautrès , Nathan Claudet , Mehdi Mhalla , Simon Perdrix , Valentin Savin , Stéphan Thomassé

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ACM Subject Classification
  • Theory of computation → Quantum information theory
  • Theory of computation → Quantum communication complexity
  • Mathematics of computing → Graph theory
Keywords
  • Quantum networks
  • graph states
  • vertex-minors
  • k-pairability

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Abstract

We study the notion of k-stabilizer universal quantum state, that is, an n-qubit quantum state, such that it is possible to induce any stabilizer state on any k qubits, by using only local operations and classical communications. These states generalize the notion of k-pairable states introduced by Bravyi et al., and can be studied from a combinatorial perspective using graph states and k-vertex-minor universal graphs. First, we demonstrate the existence of k-stabilizer universal graph states that are optimal in size with n = Θ(k²) qubits. We also provide parameters for which a random graph state on Θ(k²) qubits is k-stabilizer universal with high probability. Our second contribution consists of two explicit constructions of k-stabilizer universal graph states on n = O(k⁴) qubits. Both rely upon the incidence graph of the projective plane over a finite field 𝔽_q. This provides a major improvement over the previously known explicit construction of k-pairable graph states with n = O(2^{3k}), bringing forth a new and potentially powerful family of multipartite quantum resources.

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Maxime Cautrès, Nathan Claudet, Mehdi Mhalla, Simon Perdrix, Valentin Savin, and Stéphan Thomassé. Vertex-Minor Universal Graphs for Generating Entangled Quantum Subsystems. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 36:1-36:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICALP.2024.36

Author Details

Maxime Cautrès
  • Université Grenoble Alpes, CEA-Léti, F-38054 Grenoble, France
  • École Normale Supérieure de Lyon, F-69007 Lyon, France
Nathan Claudet
  • Inria Mocqua, LORIA, CNRS, Université de Lorraine, F-54000 Nancy, France
Mehdi Mhalla
  • Université Grenoble Alpes, CNRS, Grenoble INP, LIG, F-38000 Grenoble, France
Simon Perdrix
  • Inria Mocqua, LORIA, CNRS, Université de Lorraine, F-54000 Nancy, France
Valentin Savin
  • Université Grenoble Alpes, CEA-Léti, F-38054 Grenoble, France
Stéphan Thomassé
  • Université de Lyon, École Normale Supérieure de Lyon, UCBL, CNRS, LIP, F-69007 Lyon, France

Funding

This work was supported by the PEPR integrated project EPiQ ANR-22-PETQ-0007 part of Plan France 2030, by the STIC-AmSud project Qapla’ 21-STIC-10, by the QuantERA grant EQUIP ANR-22-QUA2-0005-01, and by the European projects NEASQC and HPCQS.

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