Zkoumání komplexních supravodivých nanohybridů pomocí kvantových stavů založených na neurálních sítích

• Thesis title in Czech: Zkoumání komplexních supravodivých nanohybridů pomocí kvantových stavů založených na neurálních sítích
• Thesis title in English: Exploring Complex Superconducting Nanohybrids with Neural Network Quantum States
• Key words: Supravodivos|Kvantové tečky|Neurální kvantové stavy|Strojové učení
• English key words: Superconductivity|Quantum Dots|Neural Network Quantum States|Machine Learning
• Academic year of topic announcement: 2024/2025
• Thesis type: diploma thesis
• Department: Department of Condensed Matter Physics (32-KFKL)
• Supervisor: RNDr. Martin Žonda, Ph.D.
• Author: hidden - assigned and confirmed by the Study Dept.
• Date of registration: 29.01.2025
• Date of assignment: 30.01.2025
• Confirmed by Study dept. on: 01.02.2025
• Advisors: RNDr. Pavel Baláž, Ph.D.

Guidelines

To successfully complete this thesis, the student will need to develop mastery in several key areas:

1.Fundamental Physics of SC Nanohybrids
Understanding the principles of superconductivity, quantum dots, and transport theory that underpin the rich properties of these hybrid structures.

2. Approximate Methods for Qualitative Insights
Becoming familiar with techniques such as the generalized superconducting atomic limit and the zero-band-width approximation to gain a conceptual grasp of SC nanohybrid behavior.

3. Machine Learning Techniques in Physics
Learning methods for data reduction, automated phase classification, and neural network architectures relevant to the study of strongly correlated systems.

4. Analytical Mapping of SC Components
Gaining proficiency in mapping superconducting leads and components onto finite or semi-infinite chain models to facilitate computational treatments.

5. Neural Network Quantum States
Developing an in-depth understanding of NNQS, especially the various versions of Restricted Boltzmann Machines (RBMs) and their role in representing many-body quantum states.

6. Proficiency with VMC and NetKet
Acquiring hands-on experience with Variational Monte Carlo simulations using NetKet, focusing on NNQS-based modeling and optimization protocols.

7. Analysis of Experimentally Relevant Systems
Applying all the above methods to study real-world SC nanohybrids, providing both theoretical insights and potential guidance for future experimental investigations.

By integrating these areas of expertise, the thesis seeks to advance the theoretical framework required to tackle increasingly complex superconducting nanohybrids, contributing to the design and understanding of next-generation quantum devices.

References

[1] V. Meden; The Anderson–Josephson quantum dot—a theory perspective. Journal of Physics: Condensed Matter, 31(16), 163001 (2019).
[2] A. Kadlecová, M. Žonda, V. Pokorný, & T. Novotný; Practical guide to quantum phase transitions in quantum-dot-based tunable Josephson junctions.” Phys. Rev. App., 11(4), 044094 (2019).
[3] P. Zalom et al.; Double quantum dot Andreev molecules: Phase diagrams and critical evaluation of effective models, Phys. Rev. B 110, 134506 (2024)
[4] Ch. Li, V. Pokorný, M. Žonda at al.; Individual Assembly of Radical Molecules on Superconductors: Demonstrating Quantum Spin Behavior and Bistable Charge Rearrangement, ACS Nano 19, (3), 3403 (2025)
[5] P. Zalom, M. Žonda, and T. Novotný; Hidden Symmetry in Interacting-Quantum-Dot-Based, Multiterminal Josephson Junctions, Phys. Rev. Lett. 132, 126505 (2024)
[6] F. Vicentini et al.; NetKet 3: Machine Learning Toolbox for Many-Body Quantum Systems, SciPost Phys. Codebases 7 (2022)
[7] M. Mezera, J. Menšíková, P. Baláž, M. Žonda; Neural network quantum states analysis of the Shastry-Sutherland model, SciPost Physics Core 6 (4), 088 (2023)
[8] J. Arnold, F. Schäfer, M. Žonda, AUJ Lode, Interpretable and unsupervised phase classification, Phys. Rev. Res. 3 (3), 033052 (2021)
[9] J. Gubernatis, N. Kawashima, P. Werner, Quantum Monte Carlo Methods, Algorithms for Lattice Models, Cambridge University Press (2016)
[10] A. Avella, F. Mancini, Strongly Correlated Systesm, Springer (2013)
[11] M. Erdmann et al., Deep Learning for Physics Research, World Scientific Publ.(2021)

Preliminary scope of work in English

Superconducting (SC) nanohybrids are emerging as key building blocks for next-generation quantum devices and engineered quantum materials. By integrating nanostructures—such as single atoms, molecules, or quantum dots—with superconducting surfaces or electrodes, these hybrid systems exhibit highly tunable properties. Even minimal setups, where a single quantum dot couples to one or two SC leads, can reveal a plethora of phenomena, including quantum phase transitions, the Kondo effect in the presence of superconductivity, and the formation of bound states. These fundamental insights have spurred the design of increasingly complex nanohybrids essential for advanced quantum technologies, such as SC transistors, diodes, qubits, and more intricate architectures that, for example, combine frustrated quantum magnets with superconductivity.

Despite the promise of SC nanohybrids, capturing their physics with high accuracy remains a formidable theoretical challenge. Straightforward approximations that work well for simpler systems often fail when confronted with the rich correlation effects and multiple degrees of freedom in more advanced designs. Moreover, the computational complexity of exact numerical methods scales exponentially with the number of interacting components. To circumvent these issues, it is crucial to employ and develop innovative computational approaches. One significant step in this direction has been the discovery of an exact mapping that transforms SC leads into effectively one-dimensional fermionic chains. This transformation enables the use of modern numerical techniques, such as Variational Monte Carlo (VMC) schemes built upon Neural Network Quantum States (NNQS).

Neural Network Quantum States represent a powerful and flexible framework for tackling challenging many-body quantum problems. In essence, NNQS use machine-learning architectures to encode the wavefunction of a quantum system in a highly expressive yet systematically improvable manner. One canonical example is the Restricted Boltzmann Machine (RBM), which introduces two layers of neurons: a “visible” layer that corresponds to the physical basis states of the system, and a “hidden” layer that captures non-trivial correlations among these states. By optimizing the parameters of the neural network through variational principles—often using advanced stochastic or gradient-based methods—NNQS can learn to approximate ground states, excited states, or even time-evolved states with remarkable accuracy. Beyond the RBM, more sophisticated neural-network architectures (e.g., feedforward, convolutional, or autoregressive models) have also been applied to capture the complex entanglement structure often present in correlated electronic systems. This versatility makes NNQS a compelling approach for studying SC nanohybrids, where strong correlations and competing interactions demand flexible and accurate numerical representations.

This diploma thesis aims to apply and further develop state-of-the-art VMC methods based on NNQS to investigate multi-quantum-dot systems coupled to multiple superconducting leads. Two recent findings strongly motivate this work. First, experiments by Dr. Chao Li and collaborators have demonstrated the possibility of assembling radical organic molecules on SC surfaces, paving the way for constructing magnetically frustrated trimers. Our previous theoretical results indicate that such systems can be engineered to exhibit intriguing frustration phenomena. Second, our investigation of multiterminal SC devices has shown that even relatively simple three-terminal setups can function as superconducting diodes and transistors. These developments suggest enormous potential for designing novel elements in quantum circuits, but they also pose significant challenges for standard theoretical methods.

Our preliminary studies show that Variational Monte Carlo, powered by NNQS, provides a controlled and tractable framework for addressing these challenges. By leveraging the exact mapping of SC leads to fermionic chains, we can systematically apply VMC-based methods and obtain converged results for both ground-state and nonequilibrium or time-dependent problems. For the practical implementation, we will use the NNQS functionalities in the open-source NetKet library, which offers flexible tools for defining custom neural-network architectures and running efficient Monte Carlo optimizations. This powerful combination will allow us not only to study the static properties of the nanohybrids but also to explore their dynamical behavior under various parameter regimes and external driving conditions.