Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Vertex-minor universal graphs for generating entangled quantum subsystems

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

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 $n=Θ(k^2)$ qubits. We also provide parameters for which a random graph state on $Θ(k^2)$ 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.

Vertex-minor universal graphs for generating entangled quantum subsystems

Abstract

We study the notion of -stabilizer universal quantum state, that is, an -qubit quantum state, such that it is possible to induce any stabilizer state on any qubits, by using only local operations and classical communications. These states generalize the notion of -pairable states introduced by Bravyi et al., and can be studied from a combinatorial perspective using graph states and -vertex-minor universal graphs. First, we demonstrate the existence of -stabilizer universal graph states that are optimal in size with qubits. We also provide parameters for which a random graph state on qubits is -stabilizer universal with high probability. Our second contribution consists of two explicit constructions of -stabilizer universal graph states on qubits. Both rely upon the incidence graph of the projective plane over a finite field . This provides a major improvement over the previously known explicit construction of -pairable graph states with , bringing forth a new and potentially powerful family of multipartite quantum resources.
Paper Structure (5 sections, 9 theorems, 5 equations, 1 figure)

This paper contains 5 sections, 9 theorems, 5 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Vertex-minor and stabilizer universality
  3. Existence of $k$-vertex-minor universal graphs of order quadratic in $k$
  4. Vertex-minor universal graphs from projective planes
  5. Sufficient conditions for $k$-pairability and $k$-vertex-minor universality

Key Result

Proposition 4

A graph $G$ is $k$-pairable if and only if the corresponding graph state ${\lvert G\rangle}$ is $k$-pairable using only local Clifford operations, local Pauli measurements, and classical communication.

Figures (1)

  • Figure 1: Implications between pairability, vertex-minor universality and stabilizer universality of graphs and graph states.

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 4
  • Proposition 5
  • Corollary 6
  • Proposition 7: claudet2023small
  • Theorem 8
  • Proposition 9
  • Lemma 10
  • ...and 3 more