Quantum network tomography of Rydberg arrays by machine learning

TL;DR

The paper addresses the challenge of identifying open quantum system models from restricted transport measurements by applying a multi-branch neural network pipeline to Rydberg atom arrays. It first classifies the number of nodes $M$ in a hidden network and then performs $M$-specific regression to locate the atoms and reconstruct the effective Hamiltonian and Lindblad operators ${\\hat H}'$ and ${\\hat L}$ from end-point transport data, including the presence of controllable decoherence. The results show high classification accuracy (e.g., $93.8\%$ with RF) and robust regression performance for atom localization and operator reconstruction across realistic system sizes, decoherence levels, and measurement constraints, with breakdown occurring when decoherence dominates the dipolar couplings. This work provides a data-driven framework for open-quantum-system identification in programmable quantum platforms and points toward extensions such as quantum state tomography and applications to more complex molecular or device networks.

Abstract

Configurable arrays of optically trapped Rydberg atoms are a versatile platform for quantum computation and quantum simulation, also allowing controllable decoherence. We demonstrate theoretically, that they also enable proof-of-principle demonstrations for a technique to build models for open quantum dynamics by machine learning with artificial neural networks, recently proposed in [Mukherjee et al. [arXiv:2409.18822] (2024)]. Using the outcome of quantum transport through a network of sites that correspond to excited Rydberg atoms, the multi-stage neural network algorithm successfully identifies the number of atoms (or nodes in the network), and subsequently their location. It further extracts an effective interaction Hamiltonian and decoherence operators induced by the environment. To probe the Rydberg array, one initiates dynamics repeatedly from the same initial state and then measures the transport probability to an output atom. Large datasets are generated by varying the position of the latter. Measurements are required in only one single basis, making the approach complementary to e.g. quantum process tomography. The cold atom platform discussed in this article can be used to explore the performance of the proposed protocol when training the neural network with simulation data, but then applying it to construct models based on experimental data.

Figures (7)

• Figure 1: Schematic of setup and multi-branch pipeline for Rydberg quantum network tomography. (a) An assembly of Rydberg atoms (large spheres), with a single Rydberg $p$-state at the input site (red sphere). The blue spheres inside the box region of side-length $L$ show atoms initially in a Rydberg $s$-state. These atoms define the quantum network to be probed, with small green circles representing background gas atoms that provide controllable decoherence. Violet spheres are the output Rydberg $s$-state atoms, the positions of which can be varied along dashed lines as sketched. Measurements of the transport probabilities $P_{1,2}$ from input to output sites for the range of positions then generate large data-sets. (b) These provide the input to a machine learning based multi-branch pipeline, which first classifies the system in the box based on the number of its network nodes $M$ and then uses multi-target regression for each class to locate the position of the atoms ($\mathbf{X}_m$) and matrix elements of key operators such as effective Hamiltonian $\hat{H}'$ and Lindblad decay operator $\hat{L}$ based on the excitation transport probabilities $P_k$. The latter steps all leverage an artificial neural network (ANN).
• Figure 2: Determination of the number of network nodes. (a) Excitation transport in a Rydberg aggregate of $N=7$ atoms ($M=4$ in the box), showing population on the input site (red-solid), output atoms (black dot-dashed, magenta dot-dashed) and atoms inside the black-box (dashed). Positions of the atoms are shown in the inset with colour matching the corresponding lines. The blue dashed vertical line indicates $t_{end} = 0.05\mu$s for which output probabilities are extracted for later use. (b) Resultant input datasets at $t_{end} = 0.05\mu$s, described by Eq. (\ref{['NN_input_datasets']}) for $M=1$ (red solid), 2 (green dot-dashed), 3 (blue dashed), and 4 (magenta dotted) for a single realisation of random atom positions in the box. The inset shows a zoom on the low probability structure. (c-e) Confusion matrix for inference of the number of nodes $M$, resulting from different classification algorithms (c) K-Nearest Neighbor (KNN), (d) Support Vector Machine (SVM) and (e) Random Forest (RF), here the label corresponds to $M$, visualized in the top-most panel. The box size of $L=10\mu$m is the same for (a)-(e).
• Figure 3: Input data sets for different black box sizes, with atoms subject to finite mean decoherence $\gamma/2\pi=10^2$ MHz. Input datasets are shown for (a) $L=10\mu$m, (b) $L=15\mu$m and (c) $L=20\mu$m similar to Fig. \ref{['NN_confusion_mat']}, but here with $\gamma>0$. (d) Accuracy of atom number classification by Random Forest with an increase in mean decoherence $\gamma$ for $L=10\mu$m (red circle), $L=15\mu$m (blue triangle) and $L=20\mu$m (green square). See appendix \ref{['app:decoh_effect_pop']} for corresponding excitation transport dynamics at $1\leq\gamma/2\pi \leq 10^3$ MHz ($\hbar=1$).
• Figure 4: Finding Rydberg locations with the neural network. (a) Mean absolute error (MAE) between predicted position and actual position for $M=1$ and box sizes $L=10\mu m$ (red circle), $L=15\mu$m (blue triangle) and $L=20\mu$m (green square). Horizontal dot-dashed lines indicate the maximum MAE for each box size in the same colors, averaged over $10^3$ random realisations, see text. (b)-(d) Mean relative error (MRE) described by Eq. (\ref{['NN_mae_set']}) for $M=2,3,4$, respectively as sketched in the insets, for box sizes as in (a). Horizontal black dot-dashed lines indicate the maximum MRE, see Eq. (\ref{['NN_mae_max']}). Vertical dashed lines indicate average dipolar interaction strengths from ${\hat{H}}_{\hbox{!! \scriptsize agg}}$, with colors allocated to box sizes as in other panels. A shared legend is provided in (d).
• Figure 5: Machine learning of system and environment properties. We show a comparison between predicted and actual matrix elements of (a) $\hat{H}'$ and (b) $\hat{L}$ defined in Eq. (\ref{['NN_Heff']}) and Eq. (\ref{['NN_Leff']}), respectively. A histogram of the distribution of the respective effective operators is shown in the inset. EIT parameters, see appendix \ref{['app:controllable_decoherence']}, are: $1\leq \Omega_p/2\pi\leq 13$ MHz, $\Omega_c/2\pi=30$ MHz, $\Gamma_p/2\pi=6.1$ MHz, $N_{bg}=6,400$, $\rho_{bg}=1.6\times 10^{13}$ m$^{-2}$ and $|\,{u}\,\rangle\equiv|\,{38s}\,\rangle$. Distribution of mean relative error (MRE) for (c) $\hat{H}'$ and (d) $\hat{L}$.