Quantum Network Tomography

TL;DR

Quantum Network Tomography (QNT) develops an end-to-end framework for characterizing noise in quantum networks by treating link noise as CPTP maps inferred from distributed quantum states. The approach leverages a state-distribution perspective and adapts quantum process tomography concepts to network settings, using metrics like the Quantum Fisher Information and the Quantum Cramér-Rao Bound to evaluate estimator performance. The paper concretely demonstrates depolarizing-star tomography via multicast protocols with $Z$-diagonal and GHZ-diagonal inputs, deriving QCRB and MSE results that reveal parameter-dependent behavior and sample-size requirements. It also outlines key challenges and directions, including imperfect memories, identifiability of general Pauli channels, topology and loss estimation, adaptive schemes, and integration with QKD and syndrome measurements, all aimed at enabling error-aware quantum networking. Overall, QNT offers a principled, polynomial-time pathway to verify and validate quantum network implementations, paving the way for robust, scalable quantum communication infrastructures.

Abstract

Errors are the fundamental barrier to the development of quantum systems. Quantum networks are complex systems formed by the interconnection of multiple components and suffer from error accumulation. Characterizing errors introduced by quantum network components becomes a fundamental task to overcome their depleting effects in quantum communication. Quantum Network Tomography (QNT) addresses end-to-end characterization of link errors in quantum networks. It is a tool for building error-aware applications, network management, and system validation. We provide an overview of QNT and its initial results for characterizing quantum star networks. We apply a previously defined QNT protocol for estimating bit-flip channels to estimate depolarizing channels. We analyze the performance of our estimators numerically by assessing the Quantum Cramèr-Rao Bound (QCRB) and the Mean Square Error (MSE) in the finite sample regime. Finally, we provide a discussion on current challenges in the field of QNT and elicit exciting research directions for future investigation.

Figures (5)

• Figure 1: QNT framework example. Two distribution circuits $C_1$ and $C_2$ are used to generate a measurement database with five components. In this example, $U_2$ represents an arbitrary 2-qubit gate. We show how qubits are placed in the star through atoms with indices matching the circuit description. Arrows indicate which qubits are transmitted through the links. Each copy of a state distributed through $C_1$ is measured either with local measurements in the $X$ basis, local measurements in the $Z$ basis, or a global measurement in a $4$-qubit entangled basis, e.g., . For $C_2$, local measurements in the $X$ and $Y$ basis are used. Each component (colored disks) of the measurement database corresponds to the combination of a state distribution circuit with a measurement operator.
• Figure 2: Multicast circuit. Root ($v_0$), intermediate node ($v_{n}$), and leaves ($v_{1}$ to $v_{n-1}$) are depicted as the shaded, dashed line, and continuous line nodes, respectively. Qubits resulting from the multicast circuit, depicted in the figure as $q_i$ for $i \in \{0, 1,\ldots, n - 2\}$, are forwarded to the leaves. The state $\ket{\psi}$ represents either the single-qubit state $\ket{0}$ or the Bell pair $(\ket{00} + \ket{11}) / \sqrt{2}$.
• Figure 3: QCRB, i.e., QFIM's inverse trace, per depolarizing parameter for $Z$- and GHZ-diagonal states. Curves show the QCRB when the Multicast protocol is used to characterize depolarizing stars with different sizes. We investigate the scenario where every channel in the network has the same depolarizing probability. There is a singularity in the estimators corresponding to the case when the depolarizing parameter is 0.75. In this regime, every link in the star outputs the maximally mixed state independent of the state used as input.
• Figure 4: Mean squared error with number of samples. Simulations used stars with uniform depolarizing probability of $0.1$ per link. We vary the number of samples from $10^{2}$ to $\times10^{4}$. Markers are placed every $10^{3}$ samples. We average results on $200$ trials for each value of the depolarizing probability.
• Figure 5: Decomposition of a tree into virtual stars. Paths $(v_0, v_1, v_2)$ and $(v_1, v_2, v_3)$ are virtual links in stars 1 and 2, respectively. Nodes $v_5$ and $v_4$ are not used in stars 1 and 2, respectively. Applying star characterization in virtual star 1 provides estimates for links $(v_2, v_3)$ and $(v_2, v_4)$ and for the virtual link $(v_0, v_1, v_2)$. In the case of virtual star 2, links $(v_0, v_1)$ and $(v_1, v_5)$ and the virtual link $(v_1, v_2, v_3)$ are characterized. Link $(v_1, v_2)$ can be estimated by triangulating parameter estimates from both stars.