Recent advances in the fabrication of superconducting nanostructures—microscopic systems that can be described by a few active orbitals coupled to a superconducting environment—raise the question of how to model these devices as their complexity grows. Effective low-energy models are a standard approach [1,2], but even these quickly become computationally intractable for exact methods. A representative example is provided by finite quantum dots or molecular clusters on superconducting substrates [3,4]. These systems are interesting both from a fundamental perspective and as potential platforms for engineered quantum materials or information-processing devices, such as superconducting quantum dot cellular automata. In practice, however, clusters with tens of quantum dots are already beyond the reach of exact numerical treatments. In this talk, I will explore how far modern many-body techniques can be pushed in this setting. I will discuss matrix product state (MPS)-based density matrix renormalization group (DMRG) methods for one-dimensional systems, and neural quantum states (NQS) with variational Monte Carlo for higher-dimensional clusters. A key challenge for NQS is that the relevant models do not conserve particle number; I will show how this can be addressed using a simple canonical transformation. I will introduce a Hubbard-like model derived as the superconducting “atomic limit” of the Anderson impurity model, and analyze selected limiting cases using analytical methods and exact diagonalization, already revealing a rich phase structure. I will then present DMRG results for longer chains, focusing on the role of entanglement, and finally show that NQS can access regimes where DMRG becomes inefficient [5].

References

[1] M. Žonda, P. Zalom, T. Novotný, G. Loukeris, J. Bätge, V. Pokorný, Phys. Rev. B 107 (11), 115407 (2023)​
[2] D. Bobok, L. Frk, V. Pokorný, M. Žonda, Phys. Rev. B 112 (20), 205418 (2025)​
[3] C. Li, V. Pokorny, M. Žonda, et al. ACS nano 19 (3), 3403-3413 (2025)​
[4] C. Li, V. Pokorný, P. Hapala, M. Žonda, et al. preprint arXiv:2508.05575 (2025)​
[5] J. Kodrlová, J. Rekas, T. Poláček, V. Pokorný, M. Friák, M. Žonda, In Preparation