Using a Recurrent Neural Network to Reconstruct Quantum Dynamics of a Superconducting Qubit from Physical Observations

At its core, quantum mechanics is a theory developed to describe fundamental observations in the spectroscopy of solids and gases. Despite these practical roots, however, quantum theory is infamous for being highly counterintuitive, largely due to its intrinsically probabilistic nature. Neural networks have recently emerged as a powerful tool that can extract nontrivial correlations in vast datasets. These networks routinely outperform state-of-the-art techniques in language translation, medical diagnosis, and image recognition. It remains to be seen if neural networks can be trained to predict stochastic quantum evolution without a priori specifying the rules of quantum theory. Here, we demonstrate that a recurrent neural network can be trained in real time to infer the individual quantum trajectories associated with the evolution of a superconducting qubit under unitary evolution, decoherence, and continuous measurement from physical observations only. The network extracts the system Hamiltonian, measurement operators, and physical parameters. It is also able to perform tomography of an unknown initial state without any prior calibration. This method has the potential to greatly simplify and enhance tasks in quantum systems such as noise characterization, parameter estimation, feedback, and optimization of quantum control.

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Quantum mechanics, despite its incredible success at describing the atomic and subatomic realm, remains deeply counterintuitive. The abstract form of its mathematical framework and inherently probabilistic nature lead to physical laws that are at odds with everyday experience. Is it possible, then, to infer probabilistic quantum dynamics without already knowing the rules of quantum theory? Here, we report that deep neural networks, instrumental at extracting hidden correlations in vast datasets, can infer the complex quantum dynamics of a single superconducting qubit by supervised learning.

We train a neutral network on an observation dataset consisting of elementary information about the preparation and measurement settings conjointly with the record of successive partial measurements performed on the quantum system. Without any prior insight into quantum physics, the neural network is able to predict the quantum trajectories followed by the quantum bit with an accuracy on par with quantum mathematical formalism. In fact, the network predictions show a significant improvement over quantum predictions owing to the fact that they do not require any separate calibration of physical parameters and do not assume any property of noise produced by the environment. Finally, by digging into the neural network predictions, one can reconstruct abstract quantum-mechanical quantities such as Hamiltonian and measurement operators, giving key insights into the deep nature of the quantum system.

Future work with this method will characterize bigger superconducting processors to catch subtle imperfections, improve feedback protocols for better quantum control of systems, and investigate how a neural network handles entanglement in a two-qubit system.

Figure 1

Recurrent neural network training from physical observations. (a) Schematic of the superconducting qubit dispersively coupled to a microwave cavity monitored by a high quantum efficiency Josephson parametric amplifier (JPA). The qubit is simultaneously driven on resonance at a Rabi rate ${\mathrm{\Omega }}_R$ and dispersively monitored with a strength ${\gamma }_{\varphi }$ near the cavity resonance frequency. (b) Data collected from the experimental system, comprising the preparation, measurement outcomes, and continuous measurement record of the qubit, are directly streamed to a RNN, which provides a prediction of the measurement outcome. The weights of the RNN are updated at each iteration through stochastic gradient descent. (c) The stochastic gradient descent aims at minimizing the cross-entropy loss function ${\mathcal{L}}_W$, which evaluates the distance between the prediction and the measurement outcome.

Figure 2

RNN prediction of the quantum evolution. (a) Blue-scale histogram of the normalized measurement records extracted from the experiment; traces plotted in color show two representative instances. (b) Red-scale histograms of RNN prediction for the measurement basis $b=X$, $Y$, and $Z$ in the driven case, beginning from $y_0=1$ in the preparation basis $a=X$. Traces plotted in color show the two representative predictions from the measurement records by the RNN (plain line) and by the stochastic master equation (dashed line). (c) Training validation; red-scale histogram of the RNN prediction leading to $p\pm \delta =0.85\pm 0.015$ at $T=2.5\mu \mathrm{s}$ indicated by the red maker. This subset of trajectories belongs to the validation subset $S_p$. (d) Comparison of the RNN prediction with the tomography. The tomography $⟨y_T{⟩}_{S_p}$ ($y$ axis) is given by the projective measurement outcome along $X$, $Y$, and $Z$ quadratures averaged over the subset of trajectories $S_p$ leading the prediction $p$ ($x$ axis). The orange dots correspond to the RNN predictions, and the blue dots correspond to the SME predictions. The average errors ${\epsilon }_{X,Y,Z}$ shown in the plots are used to quantify the agreements between the prediction and the tomography; it is calculated based on the method explained in the validation section of the main text. Inset: Histogram corresponding to the integrated voltage during the projective measurement for the quantum trajectory subset $S_p$ for $p=0.85\pm 0.015$. The threshold is used to discriminate the 0 and 1 states, leading to a tomography prediction $⟨y_T{⟩}_{S_p}=0.85$ in perfect agreement with the RNN prediction. The SME prediction indicates ${\mathcal{S}}_p$.

Figure 3

RNN prediction and retrodiction of the quantum evolution. (a) Red-scale histograms of the RNN prediction for the measurement basis $b=Y$ and $Z$ in the driven case beginning from $y_0=1$ in the preparation basis $a=Z$. Traces plotted in color show the two representative predictions from the measurement records by the RNN (plain line) and by the stochastic master equation (dashed line). (b) Blue-scale histogram of the normalized measurement records extracted from the experiment; traces plotted in color show representative instances. (c) Red-scale histograms of the RNN retrodiction for the same measurement record. Traces plotted in color show the two representative retrodictions from the measurement records by the RNN. (d) Retrodiction validation: Comparison of the RNN backward prediction with the tomography. The tomography $⟨y_0{⟩}_S$ ($y$ axis) is given by the initial projective measurement outcome along $X$, $Y$, and $Z$ quadratures averaged over the subset of trajectories $S_p$ leading to the prediction $p$ ($x$ axis). The blue dots correspond to the RNN predictions. The average errors ${\epsilon }_{X,Y,Z}^{\Leftarrow }$ shown in the plots are used to quantify the agreements between the prediction and the tomography; it is calculated based on the method explained in the validation section of the main text. (e) Red-scale histograms of smoothed RNN predictions based on the forward-backward analysis given by Eq. (5) for the same measurement records.

Figure 4

Parameter estimation of the quantum master equation and initial state tomography. (a) State estimation. Estimation of six initial state preparations (red circles) using maximum-likelihood estimation on backward RNN predictions (approximately 20 000 trajectories each) initialized from an undetermined projective measurement outcome; the circle radius gives the 95% confidence interval extracted from bootstrapping methods. (b) Distribution of the RNN predictions in the $Y$ and $Z$ measurement basis for all time. (c) Average drift of individual trajectories in the Bloch sphere: the vector map of the averaged evolution of RNN predictions in the $Y$ and $Z$ measurement basis between two consecutive time steps. This map captures the Hamiltonian evolution and the Lindbladian dissipation. (d) Average diffusion of individual trajectories in the Bloch sphere: computed vector map associated with the covariance of the prediction between two consecutive time steps in the $Y$ and $Z$ measurement basis. This map captures the measurement-induced backaction.