Quantum Network Tomography for General Topology with SPAM Errors

TL;DR

The paper tackles quantum network tomography (QNT) for general topologies and Pauli channels, introducing Mergecast combined with progressive etching to uniquely identify internal channels without requiring intermediate-state preparation. It extends the framework to SPAM errors with a two-parameter error model and derives SPAM-estimation protocols that preserve identifiability, plus a Pareto-efficient bypassable-channel protocol (BypassUnicast) for networks comprising bypassable Pauli channels. The approach is validated through NetSquid simulations showing robustness to photon loss and memory decoherence, and demonstrates practical workflows for SPAM calibration followed by channel estimation. Overall, the work significantly broadens the scope of QNT beyond star networks and lays groundwork for scalable, SPAM-aware tomography in realistic quantum networks with arbitrary topology.

Abstract

The goal of quantum network tomography (QNT) is the characterization of internal quantum channels in a quantum network from external peripheral operations. Prior research has primarily focused on star networks featuring bit-flip and depolarizing channels, leaving the broader problem -- such as QNT for networks with arbitrary topologies and general Pauli channels -- largely unexplored. Moreover, establishing channel identifiability remains a significant challenge even in simplified quantum star networks. In the first part of this paper, we introduce a novel network tomography method, termed Mergecast, in quantum networks. We demonstrate that Mergecast, together with a progressive etching procedure, enables the unique identification of all internal quantum channels in networks characterized by arbitrary topologies and Pauli channels. As a side contribution, we introduce a subclass of Pauli channels, termed bypassable Pauli channels, and propose a more efficient unicast-based tomography method, called BypassUnicast, for networks exclusively comprising these channels. In the second part, we extend our investigation to a more realistic QNT scenario that incorporates state preparation and measurement (SPAM) errors. We rigorously formulate SPAM errors in QNT, propose estimation protocols for such errors within QNT, and subsequently adapt our Mergecast approaches to handle networks affected by SPAM errors. Lastly, we conduct NetSquid-based simulations to corroborate the effectiveness of our proposed protocols in identifying internal quantum channels and estimating SPAM errors in quantum networks. In particular, we demonstrate that Mergecast maintains good performance under realistic conditions, such as photon loss and quantum memory decoherence.

Figures (16)

• Figure 1: Quantum network tomography: We study the problem of inferring the parameters of the internal Pauli channels (inside the cloud) from the state preparation and measurement operations of peripheral nodes, called monitors (square blue nodes outside the cloud). The purple arrow sequence across the channels $\mathcal{P}_{12}$, $\mathcal{P}_2$, and $\mathcal{P}_1$ indicates the "progressive etching" protocol detailed in Section \ref{['subsec:progressive-etching']}.
• Figure 2: Unicast protocol in a $3$-link star network. Figure \ref{['fig:simple-unicast']} illustrates the unicast protocol for two channels $\mathcal{P}_1$ and $\mathcal{P}_2$ in the star network in Figure \ref{['subfig:star']}. Symmetrically, one can apply the same unicast protocol to the other two pairs of channels $\{\mathcal{P}_2, \mathcal{P}_3\}$ and $\{\mathcal{P}_1, \mathcal{P}_3\}$. However, this approach leads to a sign ambiguity (i.e., not identifiable) in estimating the channel parameters, as discussed in Section \ref{['sec:qnt-identifiability']}.
• Figure 3: Mergecast in a $3$-link star network, where node C is in the center of the star topology, and nodes $\texttt{A}\xspace_1$, $\texttt{A}\xspace_2$, and B are the three peripheral nodes. This protocol allows unambiguous identification of all three channels $\mathcal{P}_1$, $\mathcal{P}_2$, and $\mathcal{P}_3$ without sign ambiguity.
• Figure 4: Channel dressing for Pauli channels: the dressings in second and third rows swap the positions of the Pauli parameters such that $q_X$ and $q_Y$ appears at the position of $q_Z$ in the first row, respectively.
• Figure 5: Mergecast for peripheral channel tomography: Figure \ref{['subfig:mergecast-peripheral']} shows the generalized Mergecast protocol to estimate the parameter of a peripheral channel $\mathcal{P}_1$ in a general network topology. Figure \ref{['subfig:general-mergecast-example']} is an example of applying Mergecast to a general network. Node labels $\texttt{A}\xspace_1$, $\texttt{A}\xspace_2$, $\texttt{B}\xspace$, and $\texttt{C}\xspace$ are the same across both subfigures.

Theorems & Definitions (6)

• Definition 1: Bypassable Pauli channel in Pauli basis
• Lemma 2
• Remark 3: Alternate interpretations of Figures \ref{['fig:unicast-spam-estimate-s']} and \ref{['fig:unicast-spam-estimate-m']}
• Definition 4: Identifiable channel
• Definition 5: Equivalent identifiable channel
• Remark 6