Released numerical code and source data

This directory contains the numerical data and Python code supporting the following paper figures:

• obs_r.pdf;
• scatter_hist-3qubits.pdf;
• scatter_hist-5qubits.pdf;
• scatter_hist-7qubits.pdf;
• scatter_hist-9qubits.pdf;
• scatter_hist-11qubits.pdf;
• scatter_hist-13qubits.pdf;
• scatter_hist-15qubits.pdf.

The remaining numerical files in the parent research directory are not part of this release because they do not support figures used in the manuscript.

Associated preprint:

Zi-Shen Li, Bujiao Wu, Xiao-Wei Li, and Man-Hong Yung,
“Barren Plateaus Beyond Observable Concentration,”
arXiv:2603.18479 [quant-ph], version 2, 19 June 2026.
https://doi.org/10.48550/arXiv.2603.18479

Directory structure

bps-beyond-oc/
├── .gitignore
├── CITATION.cff
├── README.md
├── requirements.txt
├── MANIFEST.sha256
├── code/
│   ├── generate_data.py
│   ├── generate_figures.py
│   └── legacy/
│       ├── run0715.py
│       └── plot0715.py
├── data/
│   ├── summary_statistics.csv
│   ├── csv/
│   │   ├── observables_n03.csv
│   │   ├── observables_n05.csv
│   │   ├── observables_n07.csv
│   │   ├── observables_n09.csv
│   │   ├── observables_n11.csv
│   │   ├── observables_n13.csv
│   │   └── observables_n15.csv
│   └── legacy_pickle/
│       ├── res1.pkl
│       └── res2.pkl
└── figures/
    ├── obs_r.pdf
    └── scatter_hist-{3,5,7,9,11,13,15}qubits.pdf

Data description

Each observables_nXX.csv file contains (10{,}000) samples for an (n)-qubit circuit. The principal numerical columns are:

Column	Definition	Unit
tr_g_rho_plus	(\operatorname{Tr}(g\rho_+))	dimensionless
tr_g_rho_minus	(\operatorname{Tr}(g\rho_-))	dimensionless

The circuit_id and parameter_draw_id columns reproduce the historical ordering of 100 circuit architectures with 100 parameter draws per circuit. They were reconstructed from the archived row ordering and historical loop structure.

The original binary arrays are retained in data/legacy_pickle/ solely for provenance. Pickle files should be treated as trusted local files and are not recommended as the primary reuse format. The CSV files are the public, portable source-data representation.

The system-size mapping of the historical arrays is:

Historical file	Array shape	Qubit numbers
res1.pkl	((6,10000,2))	(3,5,7,9,11,13)
res2.pkl	((1,10000,2))	(15)

No values are missing or filtered.

Reproduce the figures from the archived data

Python 3.9 or newer is recommended.

git clone https://github.com/Zi-Shen/bps-beyond-oc.git
cd bps-beyond-oc
python -m venv .venv
source .venv/bin/activate
python -m pip install -r requirements.txt
python code/generate_figures.py

The script creates regenerated_figures/ containing all eight PDFs and a fresh summary_statistics.csv.

To specify a different output directory:

python code/generate_figures.py --output-dir /path/to/output

Generate a new numerical dataset

The cleaned simulation script implements the historical protocol:

python code/generate_data.py --output-dir generated_data

The defaults are:

• (n=3,5,7,9,11,13,15) qubits;
• 100 random circuit architectures per system size;
• 100 random parameter draws per circuit;
• random single-qubit rotations selected from (R_x), (R_y), and (R_z);
• nearest-neighbour (R_{xx}) gates;
• observable (Z) on the final qubit;
• shifts of (+\pi/2) and (-\pi/2);
• random seed 20250718 for newly generated data.

The full simulation, particularly at (n=15), may require substantial computation time and memory.

Shifted-parameter convention

The archived data used the positional parameter index

index = n_qubits // 2

which corresponds to a(0,floor(n/2)) in the current circuit ordering. It does not correspond to the explicitly named parameter a(floor(n/2),floor(n/2)).

For scientific traceability, generate_data.py preserves the historical choice by default:

python code/generate_data.py --shift-mode legacy

The alternative interpretation can be investigated separately:

python code/generate_data.py --shift-mode central

Data generated with central must not be presented as identical to the archived paper data.

Provenance and reproducibility limitations

• The archived numerical arrays and figures were committed on July 18, 2025.
• The original notebook records Python 3.7.12 and Matplotlib 3.5.3.
• The original random seed was not recorded.
• Consequently, the CSV files reproduce the numerical values and figures exactly, but a fresh simulation cannot be expected to reproduce the archived samples bit-for-bit.
• The scripts in code/legacy/ are retained unchanged for historical provenance. They use working-directory-dependent paths and commented execution blocks; use the cleaned scripts for reuse.

Citation

Citation metadata are provided in CITATION.cff. If you use this code or data, please cite the associated preprint:

Zi-Shen Li, Bujiao Wu, Xiao-Wei Li, and Man-Hong Yung,
“Barren Plateaus Beyond Observable Concentration,”
arXiv:2603.18479 [quant-ph], version 2, 19 June 2026.
https://doi.org/10.48550/arXiv.2603.18479

Integrity checks

MANIFEST.sha256 contains SHA-256 checksums for all released files except the manifest itself. From this directory, verify them with:

shasum -a 256 -c MANIFEST.sha256