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QArray: a GPU-accelerated constant capacitance model simulator for large quantum dot arrays

by Barnaby van Straaten, Joseph Hickie, Lucas Schorling, Jonas Schuff, Federico Fedele, Natalia Ares

Submission summary

Authors (as registered SciPost users): Jonas Schuff
Submission information
Preprint Link: scipost_202404_00010v1  (pdf)
Code repository:
Date submitted: 2024-04-08 18:55
Submitted by: Schuff, Jonas
Submitted to: SciPost Physics Codebases
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
Approaches: Theoretical, Computational


Semiconductor quantum dot arrays are a leading architecture for the development of quantum technologies. Over the years, the constant capacitance model has served as a fundamental framework for simulating, understanding, and navigating the charge stability diagrams of small quantum dot arrays. However, while the size of the arrays keeps growing, solving the constant capacitance model becomes computationally prohibitive. This paper presents an open-source software package able to compute a 100 x 100 pixels charge stability diagram of a 16-dot array in less than a second. Smaller arrays can be simulated in milliseconds - faster than they could be measured experimentally, enabling the creation of diverse datasets for training machine learning models and the creation of digital twins that can interface with quantum dot devices in real-time. Our software package implements its core functionalities in the systems programming language Rust and the high-performance numerical computing library JAX. The Rust implementation benefits from advanced optimisations and parallelisation, enabling the users to take full advantage of multi-core processors. The JAX implementation allows for GPU acceleration.

Current status:
In refereeing

Reports on this Submission

Report 2 by Isidora Araya Day on 2024-5-14 (Invited Report)


1. Implemented useful algorithm for quantum dot array computations
2. Faster algorithm than state of art
3. Well written and overall clear paper


1. Several modules of the package are not sufficiently tested
2. Lacks package documentation
3. Unclear if and why the thresholded algorithm is always valid


## Summary

The paper introduces a Python package and two useful algorithms for finding the ground state configuration of weakly coupled quantum dots. This is a costly computation relevant to the quantum dot arrays community. This work reduces the time costs of such simulations significantly, allowing for the creation of larger data sets and data analysis.

Because the paper introduces a software package, QArray, I have additionally reviewed its code. The Github repository describes the package, indicates how to install it, and contains the package's license, automated tests, examples, and entire code.

The package has unit tests that are run systematically using Github actions. I have, however, found that several files have large untested amounts of code. This can be confirmed running "pytest --cov --cov-report=html". While some of these files contain only a few functions, some of these are important functions for the package functionality, computing ground states and charge configurations. Without tests, the package's code may become unreliable over time and its current functionality is hard to validate.

While most of the code in the repository has comments about usage and input types, several public functions lack docstrings. This can be confirmed installing ruff and running "ruff check . --select D103". Documentation may increase the package's visibility and help potential users and contributors. The examples currently in the repo contain code, but lack context to serve as documentation. Having complete documentation is a requirement for the acceptance criteria of SciPost Physics Codebases.

## Questions

1. Is the integer minimum unique? If the configurations of two dots is completely symmetric with respect to gate voltages, would the default algorithm and thresholded algorithm find both configurations or would they only find one? It is not clear to me that the thresholded algorithm always returns the correct integer number configuration.
2. Is there an intuitive way to understand why the thresholded algorithm works for any potential? Fig 2b delivers the point, but the potential is fairly specific.
3. Is it possible to combine quantum dots where some are electron dots and others are hole dots? The fact that $\vec{Q} = \pm e \vec{N}$ suggests not, because elements $\vec{N}$ are defined as integers. Is this a limitation?

Requested changes

1. I would like to ask the authors to improve and extend the package tests.
2. I would like to ask the authors to document public functions and write tutorials, not just examples. The authors could also consider writing package documentation, e.g. using
3. Not all figures are referenced in the main text, e.g Fig. 1a and 1b., and Section II would benefit from referring to Fig. 1a. Please ensure that all figures are referenced.
4. Gray and black lines in Fig. 1a are hard to distinguish. I would like to suggest using a different color or dashing one of the lines.
5. The paper contains a few typos ("Compted"), please fix these.
6. The references are not formatted, please format them.


Ask for major revision

  • validity: high
  • significance: high
  • originality: high
  • clarity: good
  • formatting: good
  • grammar: excellent

Anonymous Report 1 on 2024-4-17 (Invited Report)


1 - the article follows the lines of the newspaper
2 - the proposed method seems to offer a significant advantage over current practice
3 - comparison with experimental data is proposed
4 - the code is carefully commented and accompanied by examples.


- the code lacks a complete description of all available functionalities and the general structure of the implementation (Doxygen type)


In this paper, the authors present a code for the simulation of quantum dot arrays represented by a capacitance model in order to deduce charge stability diagrams.
Instead of a brute force search for the ground state involving the study of all possible occupied states, it is proposed to obtain a continuous minimum of the free energy, and then deduce a discrete solution by exploring the neighborhood of this continuous solution, which considerably reduces the algorithmic cost. A second method is also proposed, using a threshold to further reduce the number of configurations to be covered, offering an advantage in some cases.
The stability diagrams obtained experimentally are compared with the method and found to be in agreement. A performance study is also carried out to demonstrate the benefits of the approach. The appendices provide analytical and numerical proofs of the validity of the approximations made by the techniques.
This article is accompanied by a QArray python package whose implementation also uses Rust, a compiled language and offers a GPU implementation with Jax.
Given the editorial quality of the article, the scientific interest associated with it and the package provided, it seems to me that this article following the guidelines of the journal has its place in SciPost Physics Codebases. I would like to suggest a few comments and changes.

Requested changes

1 - Introduction : it would be interesting to have a few sentences in the introduction on the general background to the usefulness of charge stability diagrams.

2 - In Part II, it may be useful to restate the conditions for studying the system in the classical constant capacitance model approach.

3 - It seems to me that the formula for F(q_d,V_g) should also be recalled before injecting equation (3) and obtaining (4).

4 - In part IV .B. equation (10) reintroduce Vg in the arguments of F.

5 - On which configuration is the summation performed to obtain the softmin in the case of thermal enlargement? It seems to me that it refers to all the configurations around the continuous optimum, otherwise the problem back to the brute-force case. This is not very clear in the paragraph.

6 - I think it would be useful to add a few sentences on the benefits of charge sensors, optimal gates voltages and virtual gates. Perhaps a figure would be welcome here.

7 - Part VI: it seems to me that the article is rather allusive on the minimization method used (OSQP). I think this is an important point in the method (when the explicit result cannot be used) and the general principle should be explained. In addition, it doesn't say how precise the continuous optimum must be, how many iterations must be performed, or how much computing time must be devoted to this step.

8 - the origin of the error bars in the figures and the method of obtaining them should be more explicit.

9 - I am uncertain by Figure 5. Smaller values of t give a faster result. However, according to the authors, this is linked to a more approximate result. I think it would be good to be able to see on the figure the competition between calculation time and error committed than just running time.

10 - Appendix C : Reintroduce Q notation, introduce L and lambda

11 - the code lacks a complete description of all available functionalities and the general structure of the implementation (Doxygen type)

12 - I find it confusing that some rust code is in .py and not .rs. In order to see which part of the code is in rust, python or jax, this could typically appear in the documentation


Ask for minor revision

  • validity: good
  • significance: good
  • originality: good
  • clarity: good
  • formatting: reasonable
  • grammar: excellent

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