Quantum connectivity of quantum networks

Abstract

The practical utility of a quantum network depends on its ability to establish entanglement between arbitrary node pairs with quality sufficient to execute entanglement enabled tasks. This capability can be assessed globally, through aggregate performance over all node pairs, as well as locally, at the level of individual nodes. Since entanglement-based connections form a layer above the underlying physical topology, quantum connectivity is not adequately captured by classical topological connectivity metrics. To enable characterisation of the quantum connectivity at the level of the network (or its subnetworks), we introduce the quantum connectivity measure (QCM), which quantifies the average connection quality between pairs of network nodes. Further, we describe two quantities, the quantum-connected fraction (QCF) and the quantum clustering coefficient (QCC), naturally derived from the QCM, which capture important features of the functional connectivity of the quantum network at the level of the network and an individual node, respectively. These metrics of quantum connectivity depend crucially on the entanglement distribution protocol and the quantum network parameters in addition to its physical topology. We demonstrate the crucial distinction between topological and quantum connectivity, showing that even a fully connected graph can be functionally disconnected for quantum tasks if average network edge-concurrence falls below a critical threshold. These quantum connectivity metrics thus provide important tools for the design, optimization, and benchmarking of future quantum networks.

Figures (4)

• Figure 1: (color online) Illustration of quantum-enhanced connectivity. (Left) A star graph where the neighbors $\{1,2,3,4\}$ of the central node $0$ lack direct connections, resulting in a classical clustering coefficient of zero for node $0$. The edges $(0,i)$ represent quantum channels with edge-parameters $\mu_{0i}$. (Right) By performing entanglement swapping at the central node, effective, all-to-all connections (dashed lines) are established among the neighbors. This creates a fully connected subgraph (a clique), giving rise to a non-zero Quantum Clustering Coefficient.
• Figure 2: (Color online) Quantum Connectivity Measure (QCM) and Quantum-Connected Fraction (QCF) as a function of the average edge-concurrence $\bar{c}$, for a QIP threshold $\epsilon = 0.3$. Results are shown for two network topologies: fully connected (magenta) and random network (green). The considered random network has $N = 10^4$ nodes and average degree $k = 10$. For each topology, the solid, dashed, and dotted lines represent the QCF for the homogeneous case, the QCF for the inhomogeneous case ($\sigma^2 = 0.005$), and the QCM for the inhomogeneous case ($\sigma^2 = 0.005$), respectively. For the inhomogeneous case, the variance is adjusted near $\bar{c} = 0$ and $\bar{c} = 1$ to ensure that the concurrence distribution remains within $[0, 1]$.
• Figure 3: (color online) Waxman network with $N=500$ nodes uniformly distributed within a circular region of radius $1000$ km. Each edge of the network is distributed with an entangled state of concurrence 0.6. The color map indicates the Quantum Connectivity Measure (QCM) for a QIP task with threshold $\epsilon=0.3$, evaluated over smaller partitions of radius $200$ km, highlighting spatial variations in entanglement connectivity across the network.
• Figure 4: The probability mass function (PMF) of the considered random network with $N=10^4$ nodes and average degree $k=10$.