Meta-learning characteristics and dynamics of quantum systems

TL;DR

MALGO addresses rapid, data-efficient adaptation to new quantum systems by meta-learning a shared dynamical model with system-specific identifiers $\eta_i$ under outer parameters $\theta$. It extends iMODE with an adaptive learning rate and a global optimizer within a bilevel optimization framework, enabling effective learning from limited data across Hamiltonians such as $H=\Delta\sigma_x+(1-\Delta)\sigma_z$ and the Heisenberg model, with predictions assessed by $MSE$ on state trajectories. Empirically, MALGO outperforms iMODE, a vanilla transformer, and an MLP on both simulated quantum dynamics and experimental spin-qubit data, including extrapolation to unseen gate configurations. The method offers robustness and efficiency gains, with broad applicability to device tuning and high-throughput characterization in quantum technologies.

Abstract

While machine learning holds great promise for quantum technologies, most current methods focus on predicting or controlling a specific quantum system. Meta-learning approaches, however, can adapt to new systems for which little data is available, by leveraging knowledge obtained from previous data associated with similar systems. In this paper, we meta-learn dynamics and characteristics of closed and open two-level systems, as well as the Heisenberg model. Based on experimental data of a Loss-DiVincenzo spin-qubit hosted in a Ge/Si core/shell nanowire for different gate voltage configurations, we predict qubit characteristics i.e. $g$-factor and Rabi frequency using meta-learning. The algorithm we introduce improves upon previous state-of-the-art meta-learning methods for physics-based systems by introducing novel techniques such as adaptive learning rates and a global optimizer for improved robustness and increased computational efficiency. We benchmark our method against other meta-learning methods, a vanilla transformer, and a multilayer perceptron, and demonstrate improved performance.

Figures (4)

• Figure 1: Schematic representation of the meta-learning algorithm during training and adaptation (with illustrative algebraic expressions). For a given system $X^i_k$, $i$ denotes the system and $k$ the index of the element. Instead of learning a single mapping $Y_k = f_{\theta}(X_k)$, the goal is to learn a mapping $Y^i_k = f_{\theta}(X^i_k; \eta_i)$, which is parameterized by $\eta_i$. Each system is assigned an initially random parameter $\eta_i$. During training, the weights of a neural network $\theta$ and all $\eta_i$ are alternately updated using a gradient-based optimizer. We introduce a new adaptive learning rate for this optimization. $\alpha, \beta, \gamma$ are the learning rates. $\mathcal{L}$ is the loss function. During adaptation, a new parameter $\eta_j$ corresponding to an unseen system is optimized using a global optimizer, while $\theta$ remains fixed. When modeling a class of physical systems, $\theta$ can be interpreted as capturing the mathematical model, and $\eta$ are parameters for this model assigned to a specific system instance.
• Figure 2: (a) Different trajectories of two-level systems evolved by the Hamilitonian from Eq. \ref{['eq:twolevelhamil']} are represented on a Bloch sphere for $\Delta = 0.2,0.5,0.8$ in left-to-right order. The colors correspond to the different datasets, where blue, red, and black correspond to training, adaptation, and test set respectively. (b) The evolution of different $\eta_i$ during training is displayed. During the first 20 epochs, $\eta_i$ is set to random values. From epoch 21 to 200, through the optimization, all $\eta_i$ split up and reach stable relative differences. From epoch 201 on, all $\eta_i$ are frozen and only the weights of the neural network are still updated. (c) The assigned $\eta_i$ is compared with the true parameter $\Delta$ in the Hamiltonian. We show both the $\eta_i$ from the training set, as well as the assigned $\eta_i$ for the adapted systems. The monotone function suggests that the meta-learning algorithm learned an interpretable representation for all systems. The algorithm can both interpolate and extrapolate to new systems. (d) We benchmark the algorithm against three other methods for the considered quantum systems and report the infidelity of predicted versus actual quantum state on the test set. The faded points correspond to the average infidelity over all the test systems and data points for a single run.
• Figure 3: (a) For three gate voltage configurations, the mapping from barrier gate voltages $V_1$, $V_2$, and $V_3$ to $g$-factor is depicted via color-encoding. The shape of the points indicates the different sets, where circle, star, and square correspond to training, adaptation, and test set respectively. (b) We benchmark MALGO against three other methods for both the $g$-factor and $f_\text{Rabi}$ mapping. Single runs, as well as their average and standard deviation, are shown. The relationship between test loss and $g$-factor error or $f_\text{Rabi}$ error is also visualized.
• Figure 4: Test loss for varying sizes of the adaptation set.