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Task Concurrency and Compatibility in Measurement-Based Quantum Networks

Jakob Kaltoft Søndergaard, René Bødker Christensen, Petar Popovski

TL;DR

It is argued that compatibility should be used for resource state design, building the foundation for determining which task pairs the network should support with pre-shared entanglement and which require execution-time coordination.

Abstract

Measurement-Based Quantum Networks (MBQNs) rely on multipartite pre-shared entanglement resources to satisfy entanglement requests. Traditional designs optimize these resources for individual tasks, neglecting that multiple tasks may arrive concurrently and compete for the same entanglement. We introduce compatibility as a design-level metric, capturing whether concurrent tasks can be satisfied by the same entanglement resources. We define a worst-case notion of compatibility where nodes are prevented from coordinating after task arrival and illustrate why tasks may be incompatible. Furthermore, we explore compatibility extensions that account for stochastic arrivals and the capability to supplement the pre-shared entanglement with additional entanglement on-demand, and show that incompatibility differs structurally dependent on the set of concurrent tasks. We argue that compatibility should be used for resource state design, building the foundation for determining which task pairs the network should support with pre-shared entanglement and which require execution-time coordination. Numerical simulations demonstrate this potential, with $(G,1)$-compatibility achieving a 40%-55% gain in simultaneously supported tasks relative to the single-task baseline. By incorporating compatibility as a fundamental design objective, quantum networks can move beyond single-task optimization towards scalable, robust architectures that effectively balance proactive entanglement distribution and supplemental reactive coordination.

Task Concurrency and Compatibility in Measurement-Based Quantum Networks

TL;DR

It is argued that compatibility should be used for resource state design, building the foundation for determining which task pairs the network should support with pre-shared entanglement and which require execution-time coordination.

Abstract

Measurement-Based Quantum Networks (MBQNs) rely on multipartite pre-shared entanglement resources to satisfy entanglement requests. Traditional designs optimize these resources for individual tasks, neglecting that multiple tasks may arrive concurrently and compete for the same entanglement. We introduce compatibility as a design-level metric, capturing whether concurrent tasks can be satisfied by the same entanglement resources. We define a worst-case notion of compatibility where nodes are prevented from coordinating after task arrival and illustrate why tasks may be incompatible. Furthermore, we explore compatibility extensions that account for stochastic arrivals and the capability to supplement the pre-shared entanglement with additional entanglement on-demand, and show that incompatibility differs structurally dependent on the set of concurrent tasks. We argue that compatibility should be used for resource state design, building the foundation for determining which task pairs the network should support with pre-shared entanglement and which require execution-time coordination. Numerical simulations demonstrate this potential, with -compatibility achieving a 40%-55% gain in simultaneously supported tasks relative to the single-task baseline. By incorporating compatibility as a fundamental design objective, quantum networks can move beyond single-task optimization towards scalable, robust architectures that effectively balance proactive entanglement distribution and supplemental reactive coordination.
Paper Structure (20 sections, 1 theorem, 2 equations, 4 figures)

This paper contains 20 sections, 1 theorem, 2 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Quantum Networking Architecture and Resource Model
  3. Entanglement Resource Graph
  4. Allowed Operations
  5. Tasks
  6. Task Compatibility
  7. Worst-Case Compatibility
  8. Compatibility Scenarios
  9. Covering Tasks
  10. Intersecting Tasks
  11. Disjoint but Adjacent Tasks
  12. Separated Tasks
  13. Beyond Worst-Case Compatibility
  14. Resource State Redesign
  15. Satisfying Incompatible Tasks Partially
  16. ...and 5 more sections

Key Result

Proposition 1

In MBQN relying on LOCC via the repeater-path protocol to extract requested entanglement from the resource state $\ket{G}$, disjointness and separability are necessary conditions for two tasks to be simultaneously satisfiable by $\ket{G}$, that is, $G$-compatible.

Figures (4)

  • Figure 1: Minimal example illustrating limitations of single-task metrics under concurrent tasks. a) Tripartite entanglement resource shared by the network prior to task arrival. b) Bipartite entanglement between any node pair in isolation is satisfiable by the resource. c) Concurrent independent bipartite entanglements are not compatible by the resource.
  • Figure 2: Seven node 1D-cluster resource state used throughout the paper.
  • Figure 3: Compatibility examples on the resource state of Fig. \ref{['fig: Invited_Network_Setup']}. Highlighted paths represent concurrent tasks requests, solid ($T_1$) and dashed ($T_2$). a) Covering tasks. b) Intersecting tasks. c) Disjoint but adjacent tasks. d) Separated tasks.
  • Figure 4: Average number of simultaneously supported tasks as a function of network size for three different compatibility measures: baseline, worst-case compatibility, and $(G,1)$-compatibility. Shaded regions indicate the standard error of the mean over $10^4$ trials.

Theorems & Definitions (2)

  • Definition 1: Worst-Case Task Compatibility
  • Proposition 1: Necessity under LOCC