Vertex-minor universal graphs for generating entangled quantum subsystems
<|reference_start|>Vertex-minor universal graphs for generating entangled quantum subsystems: 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 $n=\Theta(k^2)$ qubits. We also provide parameters for which a random graph state on $\Theta(k^2)$ qubits is $k$-stabilizer universal with high probability. Our second contribution consists of two explicit constructions of $k$-stabilizer universal graph states on $n = O(k^4)$ qubits. Both rely upon the incidence graph of the projective plane over a finite field $\mathbb{F}_q$. 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.<|reference_end|>
@article{cautrès2024vertex-minor,
    title={Vertex-minor universal graphs for generating entangled quantum
  subsystems},
    author={Maxime Cautr`es, Nathan Claudet, Mehdi Mhalla, Simon Perdrix,
  Valentin Savin, St'ephan Thomass'e},
    journal={ICALP 2024},
    year={2024},
    doi={10.4230/LIPIcs.ICALP.2024.36},
    archivePrefix={arXiv},
    eprint={2402.06260},
    primaryClass={quant-ph cs.DM}
}