Jarosław Pawłowski and Mateusz Krawczyk of Wrocław University of Science and Technology introduce a generalizable framework for identifying effective Hamiltonians directly from experimental data in solid-state quantum systems. Their unsupervised autoencoder-based approach incorporates the S-matrix formalism into the decoder, enabling physically meaningful Hamiltonian inference without labeled Hamiltonian parameters. Through numerical experiments on a triple quantum dot chain analyzed via transport measurements, the model accurately infers Hamiltonian parameters and demonstrates generalization beyond the training domain.
Autoencoder Learns Hamiltonian Parameters from Quantum Dot Chains
Numerical experiments demonstrated automated characterization of programmable solid-state quantum simulators using a novel autoencoder approach and transport measurements performed on a triple quantum dot chain. The model accurately infers Hamiltonian parameters, a feat achieved without requiring labeled data for training, and exhibits generalization capabilities beyond the initially defined training domain. The framework’s robustness was confirmed through tests involving noisy measurements. The autoencoder successfully predicted Hamiltonian parameters even from conductance maps containing disturbances.
Specifically, the approach addresses reconstructing the effective Hamiltonian of a gated quantum dot chain from conductance maps as a function of applied gate voltages, a process involving parameters denoted as P = . After training using synthetic conductance maps, the predictor neural network can accurately predict these Hamiltonian parameters for a given conductance map. Reconstruction error, averaged across channels, pixels, and parameter variations, indicates the model’s precision in mapping measured conductance to underlying Hamiltonian values.
S-Matrix Formalism Defines Transport Measurements in QD Simulators
The framework presented uses the S-matrix formalism, a method for describing scattering processes, directly within the decoder of an autoencoder, bypassing the need for pre-labeled training data to infer the governing physics of quantum systems. This direct incorporation represents a departure from conventional machine learning approaches that typically require extensive, manually annotated datasets for accurate Hamiltonian inference.
By embedding the physics into the network’s architecture, the model can learn the relationships between measurable quantities and underlying system parameters without explicit supervision. These measurements, calculated via the Weidenmüller formula within the wide-band limit and using the S-matrix, provide a pathway to extract information about the system’s Hamiltonian.
This approach allows for automated parameter tuning, a critical step in optimizing the performance of quantum simulators. The model’s capacity to generalize beyond its initial training parameters is particularly noteworthy, exhibiting finite performance even as the parameter space expands. This robustness was further demonstrated by successfully training the autoencoder on transport data generated from Hamiltonian families different from those used during the decoder’s construction. This suggests the model isn’t simply memorizing training data but is instead identifying underlying physical principles, allowing it to extrapolate to new, unseen scenarios and still identify meaningful effective parameters.
“We further demonstrate robustness to noisy measurements,” the authors state, highlighting the model’s resilience in realistic experimental conditions. Analysis of the system’s conductance relied on reshaping the S-matrix into reflection matrices, denoted as and , which then provide the differential conductance in units of .
The researchers defined a “sweet spot” configuration of eigenstates, important for the emergence of Majorana zero modes, and detailed this analysis in Appendix An of their work. This level of detail emphasises the model’s ability to not only infer parameters but also to connect them to specific physical phenomena within the quantum dot chain. The work builds on previous Hamiltonian learning (HL) efforts in nanoscale solid-state platforms, including applications in identifying quantum nanomagnets from spectral data and assisting in the control of molecular spin qubits.
While earlier HL approaches have focused on diverse data types, such as density-of-states and moiré material patterns, this study specifically highlights the potential of transport measurements for automated parameter tuning. The researchers note that inverting measured conductance matrices to determine electrostatic potential disorder provides a natural starting point for precise inference of Hamiltonian parameters relevant to Majorana zero modes. The ability to accurately infer these parameters from transport measurements represents a step toward realizing the full potential of solid-state quantum simulators.
Rashba Quantum Dot Chain as a Kitaev Simulator Platform
The proposed Hamiltonian learning (HL) architecture incorporates a predictor neural network alongside a physics-decoder, enabling identification of effective Hamiltonian parameters without relying on pre-labeled training data. This approach diverges from typical supervised learning methods by embedding the governing physics, specifically, the S-matrix formalism, directly within the decoder component. Through simulations, the system demonstrated its capacity to accurately infer parameters defining a triple quantum dot chain, a minimal realization of a Kitaev simulator based on a Rashba chain coupled with a superconductor.
The model’s performance was evaluated using synthetic conductance maps generated from variations in key Hamiltonian parameters, local potential, Zeeman energy, pairing and Rashba spin-orbit vector, across the three quantum dots. These parameters, collectively denoted as P, were sampled from defined ranges around reference values, and the predictor network learned to reconstruct P from the resulting conductance maps.
Reconstruction error, averaged across channels and pixels within the conductance maps, served as the metric for evaluating the accuracy of the inferred parameters. The autoencoder exhibited robustness beyond the initial training domain, successfully characterizing the system even with variations in parameters outside the ranges used for training. This generalization capability suggests the model can potentially uncover unexpected physics within the system, identifying meaningful effective parameters even when the underlying Hamiltonian family differs from the one initially assumed.
The researchers tested this by sampling parameter variations, such as Δμ, which represents variations in the quantum dot potentials, and observing the reconstruction error across different ranges. The results indicate that the model maintains accuracy even when extrapolating beyond the training data, a key feature for real-world applications where precise knowledge of system parameters is often limited.
The specific system investigated consisted of a chain of three quantum dots, a configuration chosen for its complexity while remaining physically realizable. The Hamiltonian describing this chain incorporates terms for local potential, Zeeman energy, proximity-induced superconducting pairing, and Rashba spin-orbit effects, influencing the inter-dot hopping. Uniform pairing of 0.25 meV and a global Zeeman field of 0.5 meV were assumed for simplicity, while the remaining seven control parameters, , and , were allowed to vary, representing the tunable elements of the simulator.
A set of reference parameters, , served as the baseline for generating training data and evaluating the model’s performance. The model’s ability to function without pre-labeled data represents an advancement, reducing the reliance on extensive manual calibration and potentially accelerating the development of quantum technologies. The study’s findings demonstrate a pathway toward realizing automated parameter tuning in quantum dot-based simulators, enabling more precise control and optimization of these platforms for simulating complex quantum phenomena.
Unsupervised Autoencoder Architecture for Hamiltonian Inference
The autoencoder architecture uses a vision transformer (ViT) adapted to process 16-channel conductance maps, input tensors of 100×100 resolution, and ultimately outputs a vector of Hamiltonian parameters. This specific implementation uses a pretrained ViT architecture with approximately 86 million parameters, adjusted to accommodate the input data and parameter extraction process, demonstrating a scalable approach to Hamiltonian inference.
Training involved minimizing a loss function, ||G-G||^2, by comparing reconstructed conductance maps to the input maps, a process completed over 300 epochs using between 10000 and 50000 data points sampled across the parameter space. The resulting model demonstrates an ability to predict Hamiltonian parameters even when presented with noisy conductance maps, suggesting a level of robustness important for real-world applications.
Robustness to Noise and Generalization Beyond Training Data
The autoencoder framework demonstrated resilience to imperfections originating within the quantum system itself, going beyond simple noise in measurements. Specifically, the model maintained performance even when tested against Hamiltonian-level disorder, variations in the underlying physical parameters defining the quantum dot chain, a condition that introduces a mismatch between the simulated and actual systems. Researchers introduced disorder, quantified by a normalized strength parameter ranging from zero to one, to assess the impact on conductance reconstruction.
This predictable degradation is important, as it suggests the potential to quantify the degree of Hamiltonian mismatch through analysis of reconstruction error, offering a new diagnostic tool for characterizing solid-state quantum simulators. The team augmented synthetic conductance maps with both distortions mimicking charge noise and gate drift, and additive instrumental noise, to emulate realistic experimental conditions.
Initial tests with these noisy maps showed a natural degradation in prediction quality, but retraining the model with the augmented data restored performance, demonstrating an ability to adapt to imperfect data streams. The ratio of generalization volume, the range of parameters where the model performs well, to training volume was slightly larger than in a previous configuration, attributed to the structural similarity of conductance map peaks near the emergence of Majorana zero modes.
Further refinement of this generalization capability could involve a multi-head encoder, capable of predicting several candidate parameter sets and explicitly representing multiple inverse mapping branches. While the current implementation restricts predicted parameters to predefined, physically relevant ranges, this limitation could be relaxed in future work.
👉 More information
🗞 Learning Hamiltonians for solid-state quantum simulators
✍️ Jarosław Pawłowski and Mateusz Krawczyk
🧠 DOI: http://link.aps.org/doi/10.1103/jj3s-brfk
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