Performance metrics for the continuous distribution of entanglement in multi-user quantum networks

TL;DR

This work tackles the challenge of continuously distributing entanglement in multi-user quantum networks by introducing two time-dependent performance metrics, the virtual neighborhood size $v_i$ and the virtual node degree $k_i$, and optimizing protocol parameters via Pareto frontiers. The authors formalize a network model with physical topology $A$, qubit resources $r$, and stochastic entanglement processes (generation $p_{gen}$, swapping $p_{swap}$, consumption $p_{cons}$) subject to memory decoherence and cutoff fidelity constraints $F_{app}$; they study a baseline CD protocol, the Single Random Swap (SRS), under synchronous time slots. They derive analytical results for the no-swap case, show that consumption rate can dominate fidelity effects in high-generation regimes, and demonstrate node-dependent optimal parameters through a tree topology example; they further extend the framework to heterogeneous networks via Pareto optimization, providing practical regions of operation that meet QoS constraints. The contributions include: (i) two robust CD-performance metrics, (ii) analytical and numerical tools to compute steady-state performance, (iii) a Pareto-based method to balance competing node requirements in homogeneous and heterogeneous networks, and (iv) generalizable insights for large-scale quantum networks where continuous entanglement supply is essential. Overall, the paper provides a scalable methodology to assess feasibility and guide the design of continuous entanglement distribution protocols in realistic network topologies.

Abstract

Entangled states shared among distant nodes are frequently used in quantum network applications. When quantum resources are abundant, entangled states can be continuously distributed across the network, allowing nodes to consume them whenever necessary. This continuous distribution of entanglement enables quantum network applications to operate continuously while being regularly supplied with entangled states. Here, we focus on the steady-state performance analysis of protocols for continuous distribution of entanglement. We propose the virtual neighborhood size and the virtual node degree as performance metrics. We utilize the concept of Pareto optimality to formulate a multi-objective optimization problem to maximize the performance. As an example, we solve the problem for a quantum network with a tree topology. One of the main conclusions from our analysis is that the entanglement consumption rate has a greater impact on the protocol performance than the fidelity requirements. The metrics that we establish in this manuscript can be utilized to assess the feasibility of entanglement distribution protocols for large-scale quantum networks.

Figures (14)

• Figure 1: Illustration of a seven-node quantum network. The nodes are represented as gray circles, and physical channels connecting neighboring nodes are represented as gray lines. Entangled links are represented as black lines connecting two occupied qubits (small black circles). The physical topology is static, while the entangled links are continuously created, discarded, and consumed.
• Figure 2: (2,3)-tree network. Each node is represented as a gray circle and is connected to two other nodes in a lower level.
• Figure 3: Qubit addresses. Each qubit is identified by a qubit address consisting of three values $(i,j,m)$: $i$ is the node holding the qubit, $j$ is the neighboring node that can generate entanglement with that qubit, and $m$ is used to distinguish qubits with the first two indices $i$ and $j$. In this example, each node has two qubits per physical neighbor, i.e., $r=2$.
• Figure 4: Larger consumption rates decrease the virtual neighborhood size and the virtual node degree. Expected virtual neighborhood size (a) and virtual node degree (b) in the steady state in a quantum network with no swaps, with cutoff $t_\mathrm{cut}=10/p_\mathrm{cons}$ time steps, and with five qubits per node per physical channel ($r=5$). Both quantities are normalized by the physical degree of node $i$, $d_i$. The curves were calculated using (\ref{['eq.vi_noswaps_main']}) and (\ref{['eq.ki_noswaps_main']}).
• Figure 5: The virtual neighborhood size of every node cannot be maximized simultaneously. Expected virtual neighborhood size (a) and virtual node degree (b) in the steady state in a (2,3)-tree network running the SRS protocol vs the protocol parameter $q$. The value of $q$ that maximizes the virtual neighborhood size, indicated by the dotted lines, is node-dependent. The virtual node degree decreases monotonically with increasing $q$, since more links are consumed in swaps when $q$ is large. Other parameter values used in this experiment: $p_\mathrm{gen}=0.9$, $F_\mathrm{new} = 0.888$, $p_\mathrm{swap} = 1$, $r=5$, $T = 2000$ time steps, $M=4$, $p_\mathrm{cons}=p_\mathrm{gen}/4=0.225$, $F_\mathrm{app}=0.6$, $t_\mathrm{cut} = 56$ time steps (given by (\ref{['eq.cutoffcondition']})). Results obtained using a network simulation and Monte Carlo sampling with $10^6$ samples. Error bars are not shown since they are smaller than the line width -- the standard errors are below 0.003 and 0.006 for the $v_i$ and $k_i$, respectively. The standard error is defined as $2\hat{\sigma}/\sqrt{N_\mathrm{samples}}$, where $\hat{\sigma}$ is the sample standard deviation and $N_\mathrm{samples}$ is the number of samples.

Theorems & Definitions (9)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Lemma 1
• proof
• Definition 5
• Theorem 1
• proof