Self-Configuring Quantum Networks with Superposition of Trajectories

TL;DR

The paper tackles reliable quantum networking under unknown noise by introducing a self-configuring framework that coexists with coherent path superpositions. It leverages variational quantum optimization to adjust path amplitudes/phases and path-DOF corrections in a quantum-classical feedback loop, maximizing the CJ fidelity without characterizing channels. Key findings show consistent CJ-fidelity gains over all-path mixtures, with vacuum coherence playing a pivotal role, and a robust, scalable performance across two-node and multi-node networks—even when path-DOF noise is present. The work promises practical quantum networking progress by enabling noise-robust, adaptive routing in realistic hardware settings, and opens avenues for high-dimensional extensions and benchmarking-integrated warm starts.

Abstract

Quantum networks are a backbone of future quantum technologies thanks to their role in communication and scalable quantum computing. However, their performance is challenged by noise and decoherence. We propose a self-configuring approach that integrates superposed quantum paths with variational quantum optimization techniques. This allows networks to dynamically optimize the superposition of noisy paths across multiple nodes to establish high-fidelity connections between different parties. Our framework acts as a black box, capable of adapting to unknown noise without requiring characterization or benchmarking of the corresponding quantum channels. We also discuss the role of vacuum coherence, a quantum effect central to path superposition that impacts protocol performance. Additionally, we demonstrate that our approach remains beneficial even in the presence of imperfections in the generation of path superposition.

Figures (9)

• Figure 1: General setting. Two arbitrary parties in a quantum network seek to communicate. (a) Different paths can be used to connect parties Alice (A) and Bob (B). (b) These paths can also be utilized in coherent superposition to establish the connection. Our self-configuring framework employs a quantum-classical feedback loop (solid red arrows), in which a classical processor optimizes the fidelity $F$ of a quantum state transmitted from Alice to Bob by iteratively updating the parameters that control the path superposition, $\boldsymbol{\theta}_{\mathrm{c}1}$ and $\boldsymbol{\theta}_{\mathrm{c}2}$. Here we show the direct two-node configuration, with the full protocol illustrated in Fig. \ref{['fig:vqo_2_node']}. For the more general multi-node configuration, see Fig. \ref{['fig:vqo_multi_node']}.
• Figure 2: Schematic of the two-node protocol (Protocol \ref{['table:two_node_protocol_summary']}). The directed lines indicate processes involving the classical hardware. All red lines/boxes involve the VQO algorithm. The gray boxes enclosing Alice and Bob denote all protocol aspects that are accessible to them (preparation and post-processing steps respectively). Alice and Bob's nodes are denoted as solid black circles residing on the edge of their respective boxes.
• Figure 3: Role of vacuum coherence in the two-node protocol for $d$ identical paths. Assuming no path DOF noise and the vacuum interference operator is proportional to the zeroth Kraus operator, $F_{\mathrm{vio}}=\alpha^*_0 K_0$, we plot the optimal infidelity ratio $\mathcal{R}_{\mathrm{det}}$ for the deterministic variant of the protocol, together with the optimal postselection success probability $p_{\mathrm{succ}}$ and infidelity ratio $\mathcal{R}_{\mathrm{prob}}$ for the probabilistic variant of the protocol, as functions of the vacuum amplitude $\alpha_0$ in the case of (a) dephasing, (b) depolarizing or (c) amplitude damping noise for various incoherent (single-path) fidelities $F^{0}_{\mathrm{CJ}}$. The insets of (a3), (b3) and (c3) plot the thresholds $\alpha^{\mathrm{th}}_0$, below which the protocol is no longer advantageous deterministically, as functions of $F^{0}_{\mathrm{CJ}}$ (dotted lines). In the dephasing and depolarizing cases, we also show $\alpha_0$ given by a simple microscopic model (solid lines), see Eqs. \ref{['eq:dephasing_micro_vio']} and \ref{['eq:depolarizing_micro_vio']}. (In the amplitude damping case, the microscopic model always yields $\alpha_{0} = 1$; see Eq. \ref{['eq:vio_micro_full_ad']}.) Furthermore, in panels (a1), (b1) and (c1), the $\alpha^{\mathrm{th}}_0$ values corresponding to the chosen $F^{0}_{\mathrm{CJ}}$ are highlighted by vertical red lines.
• Figure 4: Two-node protocol for $d$ identical paths. The optimal $\mathcal{R}_{\mathrm{det}}$ for the deterministic variant and the optimal $p_{\mathrm{succ}}$ and $\mathcal{R}_{\mathrm{prob}}$ for the probabilistic variant are plotted as functions of $F^{0}_{\mathrm{CJ}}$ in the case of (a) dephasing, (b) depolarizing or (c) amplitude damping noise, where the vacuum interference operator $F_{\mathrm{vio}}$ is given by the microscopic model in App. \ref{['app:microscopic_special']}. We also show the effect of varying the path DOF noise strength $\sigma$; note that $\sigma=0.1$ is not shown for the probabilistic variant as it visually overlaps with $\sigma=0$. In particular, the threshold values $F^{0}_{\mathrm{CJ,th}}$ above which the protocol is no longer advantageous deterministically are plotted as functions of $\sigma$ in the insets of (a1), (b1) and (c1), with $\sigma=0.1$, $0.2$ and $0.5$ highlighted by vertical red lines in each panel and red markers in its inset.
• Figure 5: Two-node protocol for $d=2$ paths with dephasing and depolarizing noises, respectively. We plot the optimal values of (a) $\mathcal{R}_{\mathrm{det}}$ for the deterministic variant, (b) $p_{\mathrm{succ}}$ and (c) $\mathcal{R}_{\mathrm{prob}}$ for the probabilistic variant as functions of $F^{0}_{\mathrm{CJ}}$ of both channels, where the vacuum interference operators $F_{\mathrm{vio}}$ are given by the microscopic model in App. \ref{['app:microscopic_special']}. The left column corresponds to results without the path DOF noise, and the right column corresponds to $\sigma=0.1$ (deterministic) and $\sigma=0.5$ (probabilistic). In each row, the inset of the left column compares the $\sigma=0$ (solid) and $\sigma\neq 0$ (dotted/dashed) cross sections along the line $F^{0,\mathrm{depha}}_{\mathrm{CJ}}+ F^{0,\mathrm{depol}}_{\mathrm{CJ}}=1.9$ (shown in the upper right corner of every panel).