pith. sign in

arxiv: 2604.05052 · v1 · submitted 2026-04-06 · ❄️ cond-mat.mes-hall · cond-mat.dis-nn

Predicting spin-orbit coupling in hole spin qubit arrays with vision-transformer-based neural networks on a generalized Hubbard model

Jacob R. Taylor , Katharina Laubscher , Sankar Das Sarma This is my paper

Pith reviewed 2026-05-10 19:21 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.dis-nn
keywords spin-orbit couplinghole quantum dotsneural networksHubbard modelcharge stability diagramsspin qubitsmachine learningvision transformers
0
0 comments X

The pith

A vision-transformer neural network predicts effective spin-orbit coupling strength in hole quantum dot arrays from charge stability diagrams even when all other model parameters are unknown.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that vision-transformer neural networks trained on simulated charge stability diagrams can extract the spin-orbit coupling strength in 2 by 2 germanium hole quantum dot arrays. The diagrams come from a generalized Hubbard model that includes random disorder in onsite potentials, Coulomb interactions, tunneling amplitudes, and the direction and angle of SOC-induced spin rotations. The network predicts the SOC-induced spin-flip tunneling amplitudes with an R squared value of approximately 0.94 without any knowledge of the remaining Hubbard parameters. The same network simultaneously predicts those other parameters to high accuracy as well. If this holds, it would allow direct readout of the key interaction governing spin qubit performance from routine experimental data.

Core claim

We introduce a neural-network-based machine learning method to predict the effective spin-orbit coupling (SOC) strength in hole quantum dot arrays from standard charge stability diagrams. We study a 2 by 2 Ge hole quantum dot array described by a generalized spin-orbit coupled Hubbard model that incorporates random site- and bond-dependent disorder in all system parameters, including onsite potentials, Coulomb interaction strengths, interdot tunneling amplitudes, as well as the direction and angle of the SOC-induced spin rotations accompanying interdot tunneling. We train the neural network on numerically simulated charge stability diagrams from nearest-neighbor pairs of quantum dots for the

What carries the argument

Vision-transformer-based neural network trained on simulated charge stability diagrams generated from the disordered generalized spin-orbit coupled Hubbard model, with the SOC-induced spin-flip tunneling amplitudes as the primary target output.

If this is right

  • Effective SOC strength becomes extractable from standard experimental charge stability diagrams without separate knowledge of disorder or other interaction terms.
  • The same network can return estimates for onsite potentials, Coulomb strengths, and tunneling amplitudes from the identical input data.
  • Prediction accuracy holds across ranges of chemical potential and out-of-plane magnetic field used in the training simulations.
  • Automated parameter extraction becomes feasible for hole spin qubit arrays whose physics is captured by the generalized disordered Hubbard model.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could shorten experimental calibration cycles for hole qubit arrays by replacing manual multi-parameter fits with a single forward pass through the trained network.
  • If the model captures the dominant physics, the network might be fine-tuned on limited experimental data to handle additional noise sources not present in simulation.
  • The vision-transformer architecture could be tested on larger dot arrays or different host materials to check whether the same training strategy scales beyond the 2 by 2 case studied here.

Load-bearing premise

Simulated charge stability diagrams from the generalized Hubbard model with added random disorder will match the features present in real experimental diagrams from fabricated hole quantum dot arrays.

What would settle it

Directly measure the spin-flip tunneling amplitudes or effective SOC strength in a real 2 by 2 Ge hole quantum dot array via transport or spectroscopy, then compare those values to the neural network predictions obtained from the measured charge stability diagram of the same device.

Figures

Figures reproduced from arXiv: 2604.05052 by Jacob R. Taylor, Katharina Laubscher, Sankar Das Sarma.

Figure 1
Figure 1. Figure 1: FIG. 1: Diagram of a [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Example of charge stability diagrams used as [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Scatter plot showing predicted vs. expected [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Scatter plot showing predicted vs. expected model parameters when all parameters, including [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
read the original abstract

We introduce a neural-network-based machine learning method to predict the effective spin-orbit coupling (SOC) strength in hole quantum dot arrays from standard charge stability diagrams. Specifically, we study a $2\times 2$ Ge hole quantum dot array described by a generalized spin-orbit coupled Hubbard model that incorporates random site- and bond-dependent disorder in all system parameters, including onsite potentials, Coulomb interaction strengths, interdot tunneling amplitudes, as well as the direction and angle of the SOC-induced spin rotations accompanying interdot tunneling. We train the neural network on numerically simulated charge stability diagrams from nearest-neighbor pairs of quantum dots for different chemical potentials and out-of-plane magnetic fields, and show that this enables us to predict the SOC-induced spin-flip tunneling amplitudes -- and, thus, the effective SOC strength -- with high fidelity ($R^2\approx 0.94$) even when all other Hubbard model parameters are unknown. Furthermore, our neural network can also predict the other Hubbard model parameters with high fidelity, demonstrating that neural-network-based approaches can be a powerful tool for the automated characterization of hole spin qubit arrays.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a vision-transformer neural network to predict the effective spin-orbit coupling (SOC) strength in 2×2 Ge hole quantum dot arrays from simulated charge stability diagrams. The underlying model is a generalized spin-orbit-coupled Hubbard Hamiltonian that includes random site- and bond-dependent disorder in onsite potentials, Coulomb interactions, interdot tunneling, and SOC-induced spin-rotation angles. The network is trained on diagrams generated for nearest-neighbor pairs at varying chemical potentials and out-of-plane magnetic fields; the central claim is that it recovers the SOC-induced spin-flip tunneling amplitudes (and thereby the effective SOC strength) with R²≈0.94 on held-out simulated data even when all other Hubbard parameters remain unknown. The network is also reported to predict the remaining Hubbard parameters with high fidelity.

Significance. If the result holds, the work offers a concrete demonstration that a vision-transformer architecture can invert a comprehensive, disordered Hubbard simulator to extract SOC parameters from standard charge-stability data. The explicit inclusion of broad random disorder in every model parameter during training is a methodological strength that tests the network’s ability to disentangle SOC effects. This empirical mapping from simulated diagrams to microscopic parameters is reproducible in principle and could accelerate automated tuning of hole spin-qubit arrays, provided the simulator-to-experiment gap is addressed.

major comments (2)
  1. [Methods/Results] Methods/Results sections: the abstract reports R²≈0.94 on simulated test data, yet no information is supplied on the total number of generated diagrams, the train/test split ratio, regularization (dropout, weight decay, early stopping), or hyperparameter search. Without these details it is impossible to judge whether the quoted fidelity reflects genuine generalization or overfitting to the chosen forward model.
  2. [Discussion] Discussion: the central claim is framed as enabling prediction “even when all other Hubbard model parameters are unknown,” but the training and test distributions are drawn from exactly the same generative process. A quantitative assessment of performance under controlled distribution shift (e.g., different disorder variances or additional noise models) is needed to support the claim that the network will remain accurate for real devices.
minor comments (2)
  1. [Abstract] Abstract: the statement that the network predicts “the other Hubbard model parameters with high fidelity” is not quantified; reporting the corresponding R² values would make the claim precise.
  2. [Figures] Figure captions and axis labels: ensure every charge-stability diagram figure explicitly states the range of chemical potentials, magnetic fields, and disorder parameters used in the simulation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment and constructive comments, which will help strengthen the manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: [Methods/Results] Methods/Results sections: the abstract reports R²≈0.94 on simulated test data, yet no information is supplied on the total number of generated diagrams, the train/test split ratio, regularization (dropout, weight decay, early stopping), or hyperparameter search. Without these details it is impossible to judge whether the quoted fidelity reflects genuine generalization or overfitting to the chosen forward model.

    Authors: We agree that these implementation details are necessary to evaluate generalization. In the revised manuscript we will expand the Methods section to report the total number of generated charge-stability diagrams, the train/test split ratio, the regularization strategies (dropout, weight decay, early stopping), and the hyperparameter search procedure. These additions will make clear that the reported R² value is obtained under standard practices that guard against overfitting. revision: yes

  2. Referee: [Discussion] Discussion: the central claim is framed as enabling prediction “even when all other Hubbard model parameters are unknown,” but the training and test distributions are drawn from exactly the same generative process. A quantitative assessment of performance under controlled distribution shift (e.g., different disorder variances or additional noise models) is needed to support the claim that the network will remain accurate for real devices.

    Authors: We acknowledge that training and test data are drawn from the identical generative process, which is the standard evaluation protocol for simulated-data studies. The broad, random disorder already present in every Hubbard parameter during training provides a non-trivial test of the network’s ability to isolate SOC effects. Nevertheless, we agree that explicit robustness checks under distribution shift would better support claims about real-device applicability. In the revised Discussion we will add a paragraph that (i) explicitly notes the shared distribution, (ii) qualitatively discusses expected behavior under moderate increases in disorder variance or added readout noise, and (iii) identifies a controlled shift analysis as an important direction for future work. A full quantitative study lies beyond the scope of the present minor revision but can be performed once additional compute is available. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central claim is an empirical demonstration that a vision-transformer NN, trained on charge-stability diagrams generated by numerically simulating a generalized Hubbard model (with all parameters including SOC drawn from broad random distributions), recovers the SOC spin-flip tunneling amplitude to R²≈0.94 on held-out simulated diagrams. Both training and test data are produced by the identical forward model, so the reported fidelity simply shows that the network can invert the chosen simulator; it does not reduce any target quantity to a fitted parameter by construction, invoke self-citations for uniqueness or ansatz, or rename a known result. The derivation chain is therefore self-contained as a standard supervised-learning inversion task with no load-bearing circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim depends on the assumption that the generalized Hubbard model with random disorder generates representative training data and that the neural network can learn the inverse mapping without additional physical constraints.

axioms (1)
  • domain assumption The generalized spin-orbit coupled Hubbard model with random site- and bond-dependent disorder in all parameters accurately captures the relevant physics of 2x2 Ge hole quantum dot arrays.
    All training and test charge stability diagrams are generated from this model.

pith-pipeline@v0.9.0 · 5510 in / 1354 out tokens · 59307 ms · 2026-05-10T19:21:26.573505+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Large Scale Optimization of Disordered Hubbard Models through Tensor and Neural Networks

    cond-mat.mes-hall 2026-04 unverdicted novelty 6.0

    Neural networks trained on local 3x3 tensor-network charge-stability data can predict on-site disorder with high accuracy (R²>0.99) for the central dot in larger 5x5 disordered Hubbard model arrays, enabling scalable ...

Reference graph

Works this paper leans on

51 extracted references · 51 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    Scappucci, C

    G. Scappucci, C. Kloeffel, F. A. Zwanenburg, D. Loss, M. Myronov, J.-J. Zhang, S. De Franceschi, G. Katsaros, and M. Veldhorst, The germanium quantum information route, Nature Reviews Materials6, 926–943 (2020)

    work page 2020

  2. [2]

    W. I. L. Lawrie, N. W. Hendrickx, F. van Riggelen, M. Russ, L. Petit, A. Sammak, G. Scappucci, and M. Veldhorst, Spin relaxation benchmarks and individ- ual qubit addressability for holes in quantum dots, Nano Letters20, 7237–7242 (2020)

    work page 2020

  3. [3]

    N. W. Hendrickx, W. I. L. Lawrie, M. Russ, F. van Rigge- len, S. L. de Snoo, R. N. Schouten, A. Sammak, G. Scap- pucci, and M. Veldhorst, A four-qubit germanium quan- tum processor, Nature591, 580–585 (2021)

    work page 2021

  4. [4]

    W. I. L. Lawrie, M. Rimbach-Russ, F. v. Riggelen, N. W. Hendrickx, S. L. d. Snoo, A. Sammak, G. Scappucci, J. Helsen, and M. Veldhorst, Simultaneous single-qubit driving of semiconductor spin qubits at the fault-tolerant threshold, Nature Communications14, 3617 (2023)

    work page 2023

  5. [5]

    Hsiao, P

    T.-K. Hsiao, P. Cova Fariña, S. D. Oosterhout, D. Jirovec, X. Zhang, C. J. van Diepen, W. I. L. Lawrie, C.-A. Wang, A. Sammak, G. Scappucci, M. Veldhorst, E. Demler, and L. M. K. Vandersypen, Exciton trans- port in a germanium quantum dot ladder, Phys. Rev. X 14, 011048 (2024)

    work page 2024

  6. [6]

    Zhang, E

    X. Zhang, E. Morozova, M. Rimbach-Russ, D. Jirovec, T.-K. Hsiao, P. C. Fariña, C.-A. Wang, S. D. Oosterhout, A. Sammak, G. Scappucci, M. Veldhorst, and L. M. K. Vandersypen, Universal control of four singlet–triplet qubits, Nature Nanotechnology20, 209 (2024)

    work page 2024

  7. [7]

    P. C. Fariña, D. Jirovec, X. Zhang, E. Morozova, S. D. Oosterhout, S. Reale, T.-K. Hsiao, G. Scappucci, M. Veldhorst, and L. M. K. Vandersypen, Site-resolved magnon and triplon dynamics on a programmable quan- tum dot spin ladder, arXiv:2506.08663 (2025)

    work page arXiv 2025

  8. [8]

    Jirovec, P

    D. Jirovec, P. C. Fariña, S. Reale, S. D. Oosterhout, X. Zhang, S. de Snoo, A. Sammak, G. Scappucci, M. Veldhorst, and L. M. K. Vandersypen, Mitigation of exchange crosstalk in dense quantum dot arrays, Phys. Rev. Appl.24, 034051 (2025)

    work page 2025

  9. [9]

    Borsoi, N

    F. Borsoi, N. W. Hendrickx, V. John, S. Motz, F. van 6 Riggelen, A. Sammak, S. L. de Snoo, G. Scappucci, and M.Veldhorst,Sharedcontrolofa16semiconductorquan- tum dot crossbar array, Nature Nanotechnology19, 21 (2022)

    work page 2022

  10. [10]

    V. John, C. X. Yu, B. van Straaten, E. A. Rodríguez- Mena, M. Rodríguez, S. D. Oosterhout, L. E. A. Ste- houwer, G. Scappucci, M. Rimbach-Russ, S. Bosco, F. Borsoi, Y.-M. Niquet, and M. Veldhorst, Robust and localised control of a 10-spin qubit array in germanium, Nature Communications16, 10560 (2025)

    work page 2025

  11. [11]

    J. J. Dijkema, X. Zhang, A. Bardakas, D. Bouman, A. Cuzzocrea, D. van Driel, D. Girardi, L. E. A. Ste- houwer, G. Scappucci, A. M. J. Zwerver, and N. W. Hendrickx, Simultaneous operation of an 18-qubit modu- lar array in germanium, arXiv preprint arXiv:2604.01063 (2026)

    work page arXiv 2026

  12. [12]

    C. A. Stafford and S. Das Sarma, Collective coulomb blockade in an array of quantum dots: A Mott-Hubbard approach, Phys. Rev. Lett.72, 3590 (1994)

    work page 1994

  13. [13]

    Kotlyar, C

    R. Kotlyar, C. A. Stafford, and S. Das Sarma, Addi- tion spectrum, persistent current, and spin polarization in coupled quantum dot arrays: Coherence, correlation, and disorder, Phys. Rev. B58, 3989 (1998)

    work page 1998

  14. [14]

    Kotlyar, C

    R. Kotlyar, C. A. Stafford, and S. Das Sarma, Correlated charge polarization in a chain of coupled quantum dots, Phys. Rev. B58, R1746 (1998)

    work page 1998

  15. [15]

    Hensgens, T

    T. Hensgens, T. Fujita, L. Janssen, X. Li, C. Van Diepen, C. Reichl, W. Wegscheider, S. Das Sarma, and L. M. Vandersypen, Quantum simulation of a Fermi–Hubbard model using a semiconductor quantum dot array, Nature 548, 70 (2017)

    work page 2017

  16. [16]

    Chou and S

    Y.-Z. Chou and S. Das Sarma, Spin ladder quantum sim- ulators from spin-orbit coupled quantum dot spin qubits, Phys. Rev. B113, 035124 (2026)

    work page 2026

  17. [17]

    Katsaros, J

    G. Katsaros, J. Kukučka, L. Vukušić, H. Watzinger, F. Gao, T. Wang, J.-J. Zhang, and K. Held, Zero field splitting of heavy-hole states in quantum dots, Nano Let- ters20, 5201 (2020)

    work page 2020

  18. [18]

    N. W. Hendrickx, D. P. Franke, A. Sammak, G. Scap- pucci, and M. Veldhorst, Fast two-qubit logic with holes in germanium, Nature577, 487 (2020)

    work page 2020

  19. [19]

    Saez-Mollejo, D

    J. Saez-Mollejo, D. Jirovec, Y. Schell, J. Kukucka, S. Calcaterra, D. Chrastina, G. Isella, M. Rimbach-Russ, S. Bosco, and G. Katsaros, Exchange anisotropies in microwave-driven singlet-triplet qubits, Nature Commu- nications16, 3862 (2025)

    work page 2025

  20. [20]

    Bosco and M

    S. Bosco and M. Rimbach-Russ, Exchange-only spin- orbit qubits in silicon and germanium, Phys. Rev. Appl. 25, L021002 (2026)

    work page 2026

  21. [21]

    Jirovec, P

    D. Jirovec, P. M. Mutter, A. Hofmann, A. Crippa, M. Rychetsky, D. L. Craig, J. Kukucka, F. Martins, A. Ballabio, N. Ares, D. Chrastina, G. Isella, G. Burkard, and G. Katsaros, Dynamics of hole singlet-triplet qubits with largeg-factor differences, Phys. Rev. Lett.128, 126803 (2022)

    work page 2022

  22. [22]

    H. Liu, T. Zhang, K. Wang, F. Gao, G. Xu, X. Zhang, S.-X. Li, G. Cao, T. Wang, J. Zhang, X. Hu, H.-O. Li, and G.-P. Guo, Gate-tunable spin-orbit coupling in a ger- manium hole double quantum dot, Phys. Rev. Appl.17, 044052 (2022)

    work page 2022

  23. [23]

    E. G. Kelly, L. Massai, B. Hetényi, M. Pita-Vidal, A. Orekhov, C. Carlsson, I. Seidler, K. Tsoukalas, L. Sommer, M. Aldeghi, S. W. Bedell, S. Paredes, F. J. Schupp, M. Mergenthaler, A. Fuhrer, G. Salis, and P. Harvey-Collard, Identifying and mitigating errors in hole spin qubit readout, arXiv preprint arXiv:2504.06898 (2025)

    work page arXiv 2025

  24. [24]

    2510.03125

    I. Seidler, B. Hetényi, L. Sommer, L. Massai, K. Tsoukalas, E. G. Kelly, A. Orekhov, M. Aldeghi, S. W. Bedell, S. Paredes, F. J. Schupp, M. Mergenthaler, G. Salis, A. Fuhrer, and P. Harvey-Collard, Spatial uni- formity of g-tensor and spin-orbit interaction in germa- nium hole spin qubits, arXiv preprint arXiv:2510.03125 (2025)

    work page arXiv 2025

  25. [25]

    van Riggelen-Doelman, C.-A

    F. van Riggelen-Doelman, C.-A. Wang, S. L. de Snoo, W. I. L. Lawrie, N. W. Hendrickx, M. Rimbach-Russ, A. Sammak, G. Scappucci, C. Déprez, and M. Veldhorst, Coherent spin qubit shuttling through germanium quan- tum dots, Nature Communications15, 5716 (2024)

    work page 2024

  26. [26]

    Bosco, J

    S. Bosco, J. Zou, and D. Loss, High-fidelity spin qubit shuttlingvialargespin-orbitinteractions,PRXQuantum 5, 020353 (2024)

    work page 2024

  27. [27]

    An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale

    A. Dosovitskiy, L. Beyer, A. Kolesnikov, D. Weissenborn, X. Zhai, T. Unterthiner, M. Dehghani, M. Minderer, G. Heigold, S. Gelly,et al., An image is worth 16x16 words: Transformersforimagerecognitionatscale,arXiv preprint arXiv:2010.11929 (2020)

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  28. [28]

    J. R. Taylor, J. D. Sau, and S. Das Sarma, Machine learn- ing the disorder landscape of Majorana nanowires, Phys. Rev. Lett.132, 206602 (2024)

    work page 2024

  29. [29]

    J. R. Taylor and S. Das Sarma, Vision transformer based deep learning of topological indicators in Majorana nanowires, Phys. Rev. B111, 104208 (2025)

    work page 2025

  30. [30]

    Durrer, B

    R. Durrer, B. Kratochwil, J. V. Koski, A. J. Landig, C. Reichl, W. Wegscheider, T. Ihn, and E. Greplova, Automated tuning of double quantum dots into specific charge states using neural networks, Phys. Rev. Appl. 13, 054019 (2020)

    work page 2020

  31. [31]

    Schuff, M

    J. Schuff, M. J. Carballido, M. Kotzagiannidis, J. C. Calvo, M. Caselli, J. Rawling, D. L. Craig, B. van Straaten, B. Severin, F. Fedele, S. Svab, P. Cheva- lier Kwon, R. S. Eggli, T. Patlatiuk, N. Korda, D. M. Zumbühl, and N. Ares, Fully autonomous tuning of a spin qubit, Nature Electronics9, 304 (2026)

    work page 2026

  32. [32]

    J. R. Taylor and S. Das Sarma, Neural network based deep learning analysis of semiconductor quantum dot qubits for automated control, Phys. Rev. B111, 035301 (2025)

    work page 2025

  33. [33]

    Paw lowski and M

    J. Pawłowski and M. Krawczyk, Learning Hamiltoni- ans for solid-state quantum simulators, arXiv preprint arXiv:2603.02889 (2026)

    work page arXiv 2026

  34. [34]

    Martinez, J

    B. Martinez, J. C. Abadillo-Uriel, E. A. Rodríguez-Mena, and Y.-M. Niquet, Hole spin manipulation in inhomoge- neous and nonseparable electric fields, Phys. Rev. B106, 235426 (2022)

    work page 2022

  35. [35]

    Geyer, B

    S. Geyer, B. Hetényi, S. Bosco, L. C. Camenzind, R. S. Eggli, A. Fuhrer, D. Loss, R. J. Warburton, D. M. Zum- bühl, and A. V. Kuhlmann, Anisotropic exchange inter- action of two hole-spin qubits, Nature Physics20, 1152 (2024)

    work page 2024

  36. [36]

    N. W. Hendrickx, L. Massai, M. Mergenthaler, F. J. Schupp, S.Paredes, S.W.Bedell, G.Salis,andA.Fuhrer, Sweet-spot operation of a germanium hole spin qubit with highly anisotropic noise sensitivity, Nature Mate- rials23, 920 (2024)

    work page 2024

  37. [37]

    Valvo, M

    E. Valvo, M. Jakob, P. D. Vecchio, M. Rimbach-Russ, and S. Bosco, Electrically tuneable variability in germa- nium hole spin qubits, arXiv preprint arXiv:2512.12702 7 (2025)

    work page arXiv 2025

  38. [38]

    Martinez and Y.-M

    B. Martinez and Y.-M. Niquet, Variability of hole-spin qubitsinplanargermanium,Phys.Rev.Appl.25,014018 (2026)

    work page 2026

  39. [39]

    Weinberg and M

    P. Weinberg and M. Bukov, Quspin: a python package for dynamics and exact diagonalisation of quantum many body systems part I: spin chains, SciPost Physics2, 003 (2017)

    work page 2017

  40. [40]

    Pedregosa, G

    F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R.Weiss, V.Dubourg, J.Vanderplas, A.Passos, D.Cour- napeau, M. Brucher, M. Perrot, and Édouard Duchesnay, Scikit-learn: Machine learning in Python, Journal of Ma- chine Learning Research12, 2825 (2011)

    work page 2011

  41. [41]

    Winkler,Spin-Orbit Coupling Effects in Two- Dimensional Electron and Hole Systems, Vol

    R. Winkler,Spin-Orbit Coupling Effects in Two- Dimensional Electron and Hole Systems, Vol. 41 (Springer-Verlag, Berlin, Heidelberg, New York, 2003)

    work page 2003

  42. [42]

    Moriya, K

    R. Moriya, K. Sawano, Y. Hoshi, S. Masubuchi, Y. Shi- raki, A. Wild, C. Neumann, G. Abstreiter, D. Bougeard, T. Koga, and T. Machida, Cubic Rashba spin-orbit in- teraction of a two-dimensional hole gas in a strained- Ge/SiGequantum well, Phys. Rev. Lett.113, 086601 (2014)

    work page 2014

  43. [43]

    Mizokuchi, P

    R. Mizokuchi, P. Torresani, R. Maurand, Z. Zeng, Y.-M. Niquet, M. Myronov, and S. De Franceschi, Hole weak anti-localization in a strained-Ge surface quantum well, Applied Physics Letters111, 063102 (2017)

    work page 2017

  44. [44]

    Marcellina, A

    E. Marcellina, A. R. Hamilton, R. Winkler, and D. Cul- cer, Spin-orbit interactions in inversion-asymmetric two- dimensional hole systems: A variational analysis, Phys. Rev. B95, 075305 (2017)

    work page 2017

  45. [45]

    L. A. Terrazos, E. Marcellina, Z. Wang, S. N. Copper- smith, M. Friesen, A. R. Hamilton, X. Hu, B. Koiller, A. L. Saraiva, D. Culcer, and R. B. Capaz, Theory of hole-spin qubits in strained germanium quantum dots, Phys. Rev. B103, 125201 (2021)

    work page 2021

  46. [46]

    Xiong, S

    J.-X. Xiong, S. Guan, J.-W. Luo, and S.-S. Li, Orientation-dependent Rashba spin-orbit coupling of two-dimensional hole gases in semiconductor quantum wells: Linear or cubic, Phys. Rev. B105, 115303 (2022)

    work page 2022

  47. [47]

    E. A. Rodríguez-Mena, J. C. Abadillo-Uriel, G. Veste, B. Martinez, J. Li, B. Sklénard, and Y.-M. Niquet, Linear-in-momentum spin orbit interactions in planar Ge/GeSi heterostructures and spin qubits, Phys. Rev. B 108, 205416 (2023)

    work page 2023

  48. [48]

    Y. Fang, P. Philippopoulos, D. Culcer, W. A. Coish, and S. Chesi, Recent advances in hole-spin qubits, Materials for Quantum Technology3, 012003 (2023)

    work page 2023

  49. [49]

    Xiangning Chen, Chen Liang, Da Huang, Esteban Real, Kaiyuan Wang, Hieu Pham, Xuanyi Dong, Thang Luong, Cho-Jui Hsieh, Yifeng Lu, and Quoc V Le

    L. Beyer, X. Zhai, and A. Kolesnikov, Better plain vit baselines for imagenet-1k, arXiv preprint arXiv:2205.01580 (2022)

    work page arXiv 2022

  50. [50]

    Paszke, S

    A. Paszke, S. Gross, F. Massa,et al., Pytorch: An im- perative style, high-performance deep learning library, in Advances in Neural Information Processing Systems 32 (2019) pp. 8024–8035

    work page 2019

  51. [51]

    Predicting spin-orbit coupling in Ge hole spin qubit arrays with vision-transformer-based neural networks

    P. W. (lucidrains), vit-pytorch,https://codeberg.org/ lucidrains/vit-pytorch(2021). 1 Supplementary Material for “Predicting spin-orbit coupling in Ge hole spin qubit arrays with vision-transformer-based neural networks” Appendix I: Extended neural network details The neural network consists of three layers of 3D trans- formers, where the input is embedde...

    work page 2021