Automated electrostatic characterization of quantum dot devices in single- and bilayer heterostructures

TL;DR

The paper addresses the challenge of scalable, automated electrostatic characterization of quantum-dot devices from charge stability diagrams (CSDs). It develops a three-phase workflow that combines ML-based transition detection (U-Nets), geometric reconstruction of transition networks, and electrostatic inference within a constant-capacitance framework to extract lever arms, charging and mutual voltages, and gate–dot capacitances for planar and bilayer Ge QD devices. The approach is validated against manual labeling and CSD simulations, recovering consistent capacitance matrices and interlayer couplings, and is shown to extend to device-level characterization by tracking transition lines as additional gates are swept. This device-agnostic method enables rapid, robust characterization across large data sets, facilitating autotuning and scalable integration of QD qubit architectures while highlighting the method’s limits where constant-capacitance may break down in strong-coupling regimes.

Abstract

As quantum dot (QD)-based spin qubits advance toward larger, more complex device architectures, rapid, automated device characterization and data analysis tools become critical. The orientation and spacing of transition lines in a charge stability diagram (CSD) contain a fingerprint of a QD device's capacitive environment, making these measurements useful tools for device characterization. However, manually interpreting these features is time-consuming, error-prone, and impractical at scale. Here, we present an automated protocol for extracting underlying capacitive properties from CSDs. Our method integrates machine learning, image processing, and object detection to identify and track charge transitions across large datasets without manual labeling. We demonstrate this method using experimentally measured data from a strained-germanium single-quantum-well (planar) and a strained-germanium double-quantum-well (bilayer) QD device. Unlike for planar QD devices, CSDs in bilayer germanium heterostructure exhibit a larger set of transitions, including interlayer tunneling and distinct loading lines for the vertically stacked QDs, making them a powerful testbed for automation methods. By analyzing the properties of many CSDs, we can statistically estimate physically relevant quantities, like relative lever arms and capacitive couplings. Thus, our protocol enables rapid extraction of useful, nontrivial information about QD devices.

Figures (19)

• Figure 1: Problem visualization for planar and bilayer QD device. (a) Schematic heterostructure for a planar germanium hole device. (b) Sample CSD from a planar germanium device, adapted from Ref. Rao24-MAViS, with left (blue), right (yellow), and interdot (green) transitions highlighted. This CSD is taken in the normalized plungers space. (c) Schematic bilayer germanium heterostructure, with two QDs formed under gate $\mathrm{P}_2$ and one QD formed under gate $\mathrm{P}_1$. (d) Example CSD taken for the bilayer system outlined in (c), with left upper (blue), left lower (red), interlayer (purple), right (yellow), and left-right interdot (green) transitions labeled. This CSD is taken in the orthogonalized plungers space, where $\mathrm{O}_1$ and $\mathrm{O}_2$ are orthogonalized versions of the plungers $\mathrm{P}_1$ and $\mathrm{P}_2$ shown in (e). (e) A schematic illustration of the bilayer device used in this work, with the sensor $\mathrm{P}_S$, plungers $\mathrm{P}_2$ and $\mathrm{P}_1$, and other relevant gates labeled. The part of the device not used in experiments is grayed out.
• Figure 2: The workflow of the ML-enabled CSD analysis. A CSD typical of a (a-i) planar double-QD system and (c-i) a bilayer triple-QD system. A CSD gradient, illustrated in (a-ii) for a planar device and in (c-ii) for a bilayer CSD, is passed through three U-Net pixel classifiers. The outputs of each classifier are shown in (b) and (d), with high-intensity regions corresponding to pixels assigned a high probability of belonging to a given transition class. The final transitions are highlighted with the bounding boxes computed using a combination of thresholding and clustering.
• Figure 3: Fine-grain classifications of the vertical lines in bilayer devices. (a) A motion of two cells across five frames from a measurement series of bilayer CSDs. Pink bounding boxes in each cell represent the transitions detected by the $\mathcal{M}_v$ model. Manually assigned LU (blue), LL (red), and LI (purple) labels for transition within the highlighted cells are shown in the insets. (b) The clustered $x$- and $y$-coordinates of the interdot locations for the measurement series shown in (a). (c) The change in the $x$-coordinates for transitions detected by the $\mathcal{M}_v$ model within the cells highlighted in (a). Colors indicate the fine-grain labels, while gray dashed lines indicate the 3-linear model fitted for each cell.
• Figure 4: Horizontal and vertical line transition in planar CSD. (a) A right (h) and left (v) loading transition line for a sample planar double-QD CSD with colored bounding boxes, each encapsulated by the corresponding cell (black dashed boxes). (b) Hough analysis of the transition line slopes highlighted in (a). Gray lines indicate the function $\tilde{H}_\mathrm{sq}$, while dashed colored lines indicate the Cauchy fits. Gray dotted lines indicate the best-fit angle $\theta_\mathrm{max}$.
• Figure 5: Transition line slopes and relative lever arms for the planar and bilayer CSDs. Schematic CSDs highlighting all relevant transition angles for an (a) planar and (e) bilayer CSD. Distributions of transition angles for a sample CSD series for the (b) planar and (f) bilayer CSD. The two histograms compare angles extracted from manually labeled CSDs (left) with angles from automatically characterized CSDs (right). Dashed lines indicate median angles. Transition angle distributions across the (c) 12 planar and (g) 8 bilayer CSD series. For each series, the points indicate medians, and the error bars represent the 10-90 percentile range. The relative lever arms for each dot-gate pair in the (d) planar and (h) bilayer QD systems, computed from the median angles in (c) and (g), respectively. For the planar CSDs analysis, we fix $\alpha_{\mathrm{N}_L : \mathrm{L}}{=}1$, and for the bilayer CSDs analysis $\alpha_{\mathrm{O}_2 : \mathrm{LU}}{=}1$, as indicated with small circles. The remaining lever arms are computed relative to these values. In (c), (d), (g), and (h), circles (triangles) indicate results for automatically characterized (manually labeled) datasets; error bars on the right indicate the 10-90 percentile range of the medians of the automatically characterized CSD series. In (c) and (d), the two manually labeled datasets correspond to the first two automated datasets on the left. In (d) and (h), gray lines connect the manually labeled CSD series with the corresponding automatically characterized series.