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A Control Architecture for Entanglement Generation Switches in Quantum Networks

Scarlett Gauthier, Gayane Vardoyan, Stephanie Wehner

TL;DR

The paper addresses scalable entanglement generation in quantum networks by introducing a central Entanglement Generation Switch (EGS) that shares heralding resources among many node pairs. It formulates resource allocation as a Network Utility Maximization (NUM) problem and derives the Rate Control Protocol (RCP) using Lagrangian duality, yielding a distributed rate-price mechanism that drives the system toward the EGS capacity region $\mathcal{C}$ with throughput-optimal scheduling via Maximum Weight Scheduling. The authors prove a capacity region theorem and a convergence theorem, and validate the approach numerically, showing robustness to dynamic changes in resources and network size. The work offers a cost-effective, scalable control architecture for quantum hubs and lays groundwork for extending similar strategies to broader quantum network models and hubs.

Abstract

Entanglement between quantum network nodes is often produced using intermediary devices - such as heralding stations - as a resource. When scaling quantum networks to many nodes, requiring a dedicated intermediary device for every pair of nodes introduces high costs. Here, we propose a cost-effective architecture to connect many quantum network nodes via a central quantum network hub called an Entanglement Generation Switch (EGS). The EGS allows multiple quantum nodes to be connected at a fixed resource cost, by sharing the resources needed to make entanglement. We propose an algorithm called the Rate Control Protocol (RCP) which moderates the level of competition for access to the hub's resources between sets of users. We proceed to prove a convergence theorem for rates yielded by the algorithm. To derive the algorithm we work in the framework of Network Utility Maximization (NUM) and make use of the theory of Lagrange multipliers and Lagrangian duality. Our EGS architecture lays the groundwork for developing control architectures compatible with other types of quantum network hubs as well as system models of greater complexity.

A Control Architecture for Entanglement Generation Switches in Quantum Networks

TL;DR

The paper addresses scalable entanglement generation in quantum networks by introducing a central Entanglement Generation Switch (EGS) that shares heralding resources among many node pairs. It formulates resource allocation as a Network Utility Maximization (NUM) problem and derives the Rate Control Protocol (RCP) using Lagrangian duality, yielding a distributed rate-price mechanism that drives the system toward the EGS capacity region with throughput-optimal scheduling via Maximum Weight Scheduling. The authors prove a capacity region theorem and a convergence theorem, and validate the approach numerically, showing robustness to dynamic changes in resources and network size. The work offers a cost-effective, scalable control architecture for quantum hubs and lays groundwork for extending similar strategies to broader quantum network models and hubs.

Abstract

Entanglement between quantum network nodes is often produced using intermediary devices - such as heralding stations - as a resource. When scaling quantum networks to many nodes, requiring a dedicated intermediary device for every pair of nodes introduces high costs. Here, we propose a cost-effective architecture to connect many quantum network nodes via a central quantum network hub called an Entanglement Generation Switch (EGS). The EGS allows multiple quantum nodes to be connected at a fixed resource cost, by sharing the resources needed to make entanglement. We propose an algorithm called the Rate Control Protocol (RCP) which moderates the level of competition for access to the hub's resources between sets of users. We proceed to prove a convergence theorem for rates yielded by the algorithm. To derive the algorithm we work in the framework of Network Utility Maximization (NUM) and make use of the theory of Lagrange multipliers and Lagrangian duality. Our EGS architecture lays the groundwork for developing control architectures compatible with other types of quantum network hubs as well as system models of greater complexity.
Paper Structure (16 sections, 10 theorems, 67 equations, 4 figures, 1 table)

This paper contains 16 sections, 10 theorems, 67 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Results Summary
  3. Related Work
  4. Preliminaries
  5. Demands for resource allocation from nodes to the EGS processor
  6. Processing demands for resource allocation
  7. Constraints
  8. RCP Algorithm
  9. Derivation
  10. Convergence
  11. Case Study
  12. Discussion
  13. Proofs
  14. Outline of goals to prove
  15. Proof of Theorem \ref{['thm:CapReg']}
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $x_{s}$ be the maximum number of resources that can be allocated to a session $s$ per time slot. For each resource, $p_{\emph{gen}}$ is the probability that a communication session allocated the resource for one time slot will successfully create an entangled pair. The capacity region of an EGS $\lambda_{\text{EGS}} = R \cdot p_{\text{gen}}$ and $\lambda^{\max}_{\text{gen}, s} = x_{s} \cdot p

Figures (4)

  • Figure 1: EGS Architecture: a) EGS structure: An EGS with $R=4$ resources connected to $N=9$ nodes. The EGS is controlled by a classical processor and consists of a switch, resources, and physical connections. Nodes have quantum communication channels to the switch and classical communication channels to the processor. b) Resource Allocation: The switch opens connections to link nodes 1, 2 and resource 1. For example, the connections may consist of direct optical fiber paths from the nodes to the switch and from the switch to the resource, via an interface at the switch. This establishes the physical allocation of resource 1 to the communication session of nodes 1, 2 for time slot $t_n$. c) Quantum communication sequence: Node-to-processor communication in time slot $t_n$ with a batch size of three entanglement generation attempts. d) Concurrent classical communication sequences: Nodes and the processor communicate in time slot $t_n$, governing resource allocation and the RCP (see Algorithm \ref{['Box:rateControlProt']} for RCP details.)
  • Figure 2: The RCP drives the sum of the demanded rates of entanglement generation across all communication sessions, $\underset{s}{\Sigma} \lambda_s(t_n)$, to converge with respect to the sequence of time slots to the maximum average entanglement generation rate of the EGS, $\lambda_{\text{EGS}}$. The EGS has $R=3$ resources, the probability of entanglement generation is $p_{\text{gen}} = 0.05$, and the EGS is connected to $N=20$ (top), $N=50$ (middle) and $N=100$ (bottom) nodes. The total number of communication sessions served are $|S| = 19, \ 123, \ \text{ and } 495$ in the top, middle, and bottom plots, respectively. Black dotted lines indicate the convergence times, $\Delta \tau$. The observed values for the tightness of convergence, $\delta$, are $\delta_1 = 0.12$, $\delta_2= 0.035$ and $\delta_3 = 0.012$. Step-sizes $(\theta_c, \theta_u \ \forall u)$ were all $1/(40 \cdot \lambda_{\text{EGS}})$.
  • Figure 3: In response to changes in the number of resources available at the EGS ($R \rightarrow R'$), the RCP drives the sum of the demanded rates of entanglement generation across all communication sessions, $\underset{s}{\Sigma} \lambda_s (t_n)$, to converge with respect to the sequence of time slots to the updated maximum average entanglement generation rate of the EGS, $\lambda_{\text{EGS}} = R' \cdot p_{\text{gen}}$. In simulation, an EGS connected to $N=50$ nodes, serving $|S|=123$ communication sessions, is originally equipped with $R=3$ resources. After every $10,000$ time-slots, one of the resources may either be taken offline for calibration or an offline node may be returned to service. Black dashed lines indicate the convergence, $\Delta \tau$ calculated for every $R'$ (initially $R$). We observe and overall tightness of convergence of $\delta=0.035$, identical to that observed in Fig. \ref{['fig:avReqRates']} for the EGS operated with fixed $R=3$ and with the same $N, \ |S|$. Step-sizes $(\theta_c, \theta_u \ \forall u)$ were all $1/(10 \cdot \lambda_{\text{EGS}})$.
  • Figure 4: Differences between the average maximum rate and average minimum rate requested by any communication session in time-slot $t_n$, for an EGS connected to $N=20$ (top), $N=50$ (middle) and $N=100$ (bottom) nodes serving $|S| = 19, \ 123, \text{ and } 495$ communication sessions, respectively. As described in the main text, nodes are either associated with a uniform and effectively un-restricted set of capabilities or a non-uniform and more restricted set of capabilities. Step-sizes $(\theta_c, \theta_u \ \forall u)$ were all $1/(40 \cdot \lambda_{\text{EGS}})$.

Theorems & Definitions (22)

  • Definition 2.1: Target Rate, Communication Session
  • Definition 2.2: Demand
  • Definition 2.3: Designated Communication Node, Secondary Node
  • Definition 2.4: Queue
  • Definition 2.5: (Demand-Based) Schedule
  • Definition 2.6: Maximum Weight Scheduling
  • Definition 2.7: Supportable rate
  • Definition 2.8: Capacity Region
  • Theorem 2.1: Capacity Region
  • Theorem 3.1: RCP Convergence
  • ...and 12 more