An Optimization Framework for Monitor Placement in Quantum Network Tomography

TL;DR

This work analyses multi-monitor configurations and shows that distributing monitors across end nodes can achieve estimation performance comparable to a monitor placed at the hub, and develops two Integer Linear Program (ILP) formulations: one maximizing estimation accuracy (QF), and another jointly optimizing accuracy and monitoring overhead (QMF).

Abstract

Quantum Network Tomography (QNT) offers a framework for end-to-end quantum channel characterization by strategically placing monitor nodes within the network. Building upon prior work on single-monitor placement, we study optimal monitor placement and measurement assignments for channel parameter estimation in arbitrary quantum networks. Using an n-node star network as a baseline, we analyze multi-monitor configurations and show that distributing monitors across end nodes can achieve estimation performance comparable to a monitor placed at the hub. Estimation precision is quantified using the Quantum Fisher Information Matrix (QFIM), with channel parameters inferred via Maximum Likelihood Estimation (MLE) and benchmarked against the Quantum Cramer-Rao Bound (QCRB). To generalize, we develop two Integer Linear Program (ILP) formulations: one maximizing estimation accuracy (QF), and another jointly optimizing accuracy and monitoring overhead (QMF). Unlike QF, QMF prevents monitor overloading, enabling scalability and parallelism. We prove optimality for star and analyze applicability to tree-structured quantum networks.

Figures (6)

• Figure 1: Probe-state distribution and measurements: Link $e_0$ is directly monitored by monitor node $v_0$, which is an endpoint of the link. Two entangled states are distributed between nodes $v_0$ and $v_1$, shown by solid red arrows. In contrast, link $e_7$ is indirectly monitored via monitor node $v_9$, as it is not incident to any monitor node. In this case, two entangled states are distributed between node pairs $(v_8, v_7)$ and $(v_7, v_9)$, indicated by dashed red arrows.
• Figure 2: Monitor placement results for a 10-node star network with nine available monitors. The x-axis indicates the number of deployed monitors from 1 to 9, and the y-axis lists the network links in descending order of their corresponding Werner parameter values. Indirectly measured links are shown in white, while directly measured links are represented in light gray. Distinct colors within each cell represent the specific monitor responsible for the measurement. Subfigures illustrate results from (a) QMF, (b) QF.
• Figure 3: Comparison of QMF and QF for the Star Network: (a) Both formulations under Heterogeneous noise, (b) Both formulations under uniform noise and estimator value set to 0.9.
• Figure 4: Monitor placement results for the 10-node tree network with up to four monitors. Node colors indicate monitor placements, and edge colors reflect the monitor performing the measurement. Solid edges denote links that are directly measured, while dashed edges represent indirect measurements. All links in the tree can be directly measured using four monitors, represented by four distinct colors, with each edge color corresponding to the monitor (of the same color) responsible for measuring that link. Subfigures show results from (a) QMF (b) QF.
• Figure 5: Numerical Analysis using QFIM: (a) QCRB of the three estimators and QCRB trace, with error rate = 0.1 in two-monitor configuration, (b) QCRB of three estimators and trace of QCRB in Three-Monitor configuration for error rate = 0.1.

Theorems & Definitions (13)

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