[WIP] Expressivity of Quantum model in terms of Fourier series.

Completed

1. Entire Report
2. Codebase for replication of every scenarios of the paper [1]. It works for the degree 1 Fourier Series
3. Replicate the figure 3 from the paper
4. Separate the report and the codes into two python notebooks and two corresponsding pdf files
5. Replicate for degree 5 Fourier series
6. Use custom 3 qubit system like the GHZ-state instead of StronglyEntanglingLayers that authors have used in their git repo

TODO(s)

1. Write OOP style code to eliminate code repetition

This repository provides an extensive and comprehensive study/review to the article [1]. Albeit, this work only covers the section I and section II, they are extensive nevertheless. I‘ve invested multiple hours and a meticulous efforts to simplify the loaded mathematical notions and symbols to make it more palatable to the reader. Since you are here, I presume you are looking to break in to the article. I hope my work will help you on your way.

The article is study about the effects of data encoding on the expressive power of variational quantum machine learning models. The main argument is the encoding parameters like number of gates and scaling of the frequency in trainable model determines the expressivity. This ultimately means the trainability of the parameteric circuit. The underlying notion to begin with is that the quantum model can be exchanged in terms of the Fourier series.

A prior knowledge of the Fourier analysis [2] is mandatory to learn from the article. Plus I would suggest the eigen-decomposition is a cool insights [3].

In the abstract section of the article the authors succinctly provide the objectives of their hard work.

Here we investigate how the strategy with which data is encoded into the model influences the expressive power of parametrised quantum circuits as function approximators. We show that one can naturally write a quantum model as a partial Fourier series in the data, where the accessible frequencies are determined by the nature of the data encoding gates in the circuit. By repeating simple data encoding gates multiple times, quantum models can access increasingly rich frequency spectra. We show that there exist quantum models which can realise all possible sets of Fourier coefficients, and therefore, if the accessible frequency spectrum is asymptotically rich enough, such models are universal function approximators.

Folder Structure

1. The folder "reports" contains the pdf file format.

• codes.pdf is the python codes which replicates the findings of the article
• fs-pl.pdf is the report of my literature review.

2. The folder "results" contains the ouput from the code in the python notebook.

Results

I was successful to replicate the results in the articles. In the file reports/codes.pdf, the Sections 1 and 2, and their subsections show the results.

After I finished replicating the “Expressivity of Quantum model for the purpose of QML” from the ref [1], I seasoned my codes with "Entanglement". Basically the article says, the parametric quantum model with n-qubits is trainable up to n-degree truncated Fourier series and the scaling of data reduces the trainability.

The authors put up StronglyEntangled 3-qubit layers to train it to 1-degree truncated Fourier series. However, I put up the on-demand 3-qubit GHZ! And it fits the 1-degree target function.

Moreover, I had an idea to check whether 3-qubit GHZ is trainable on 3-degree target function. Which it should, except if it was not for the “Entanglement” part. In my codes I found out, since the 3-qubit GHZ system is entangled the quantum model is only trainable for 1-degree target function. The 3-qubit GHZ system is untrainable for 3-degree target function.

My results (replications of the article cited herein) shows entanglement of n-qubit system makes it act like 1-qubit system!

I invite you to visit the pdf format of ipynb file in my GitHub repo and navigate to Section 3, named as, “Entangled qubits”

Quantum world is so fascinating!

Closing note

If you find this repository helpful please consider to fork or star it. Thank you for your visit!

References

[1] Schuld, Maria, Ryan Sweke, and Johannes Jakob Meyer. "Effect of data encoding on the expressive power of variational quantum-machine-learning models." Physical Review A 103.3 (2021): 032430.

[2] Weisstein, Eric W. "Fourier Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FourierSeries.html

[3] Weisstein, Eric W. "Eigen Decomposition." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EigenDecomposition.html

[4] Schuld, M., Sweke, R. and Meyer, J., 2022. GitHub - XanaduAI/expressive_power_of_quantum_models. [online] GitHub. Available at: https://github.com/XanaduAI/expressive_power_of_quantum_models [Accessed 18 February 2022]