thanatchon36/bp-from-scratch

Barren Plateaus from Scratch: results

Every data point behind the technical report Barren Plateaus from Scratch: A NumPy State-Vector Study of Trainability in Variational Quantum Circuits (Thanatchon Chongmankhong, 2026). The numbers were produced with qsim, a from-scratch NumPy state-vector simulator with exact adjoint gradients, on 2–16 qubits.

• 📄 Paper: read online · PDF
• 💻 Code: github.com/thanatchon36/bp-from-scratch (MIT)
• 📦 Archive: doi:10.5281/zenodo.23995487

The dataset comes in two forms:

• data/: one tidy Parquet table per experiment. These are the configurations below, readable with datasets, pandas or the dataset viewer.
• raw/: the JSON cache exactly as the experiment scripts wrote it, one file per configuration, plus summary.json. It is byte-identical to results/ in the code repository at v0.2.0.

Load

from datasets import load_dataset

e1 = load_dataset("thanatchon36/bp-from-scratch", "e1_variance_vs_n", split="train").to_pandas()
print(e1[["n", "L", "var_mid", "haar_two_sided"]].sort_values(["L", "n"]).head())

import pandas as pd

df = pd.read_parquet("hf://datasets/thanatchon36/bp-from-scratch/data/e5_training.parquet")
print(df.groupby(["cost", "shots", "n"]).fidelity_final.mean().unstack("n").round(3))

To regenerate every figure and table of the paper from this data, put raw/ in place of results/ in the code repository and run make figures && make paper.

Configurations

Config	Rows	One row is	What it measures
e0_haar_check	14	Hamiltonian × n	Sampled gradient variance with Haar-random circuits before, after or on both sides of a rotation, against the closed-form values (Appendix A)
e1_variance_vs_n	40	n × depth L	McClean random-axis hardware-efficient ansatz, H = Z⊗n: gradient variance, raw derivative samples and half-chain entropy
e2_local_vs_global	60	cost × n × L	Alternating-layered ansatz with a global projector cost and a local cost (Cerezo et al.)
e3_initialisation	30	strategy × n	Uniform, Gaussian (σ = 1, 0.3, 0.1) and identity-block initialisation, local cost (1/n)ΣXᵢ
e4_variance_vs_depth	13	depth L at n = 12	Fine depth grid of the E1 circuit: gradient variance and entanglement entropy
e5_training	330	one Adam run	Preparing |0⟩⊗n from random angles with exact gradients or 10⁴-shot noise; full cost history
e6_qaoa_variance	28	n × p	QAOA MaxCut on random 3-regular graphs: variance of ∂C/∂γ₁ and ∂C/∂β₁
e6_qaoa_training	18	p × graph at n = 10	Approximation ratios reached by Adam from 4 random starts per graph
e7_dla	24	family × n	Exact dynamical Lie algebra dimension, 𝔤-purities, predicted and measured Var[C]

Columns

Gradient statistics (e1–e4). Every row samples n_samples random parameter vectors and computes the full gradient with the adjoint method.

Column	Meaning
n, L	Qubits and circuit depth (layers)
n_params	Number of circuit parameters
var_first, var_mid, var_last	Variance of the derivative with respect to a probe parameter in the first, middle (middle qubit) and last layer
*_ci	95% percentile-bootstrap interval of that variance (400 resamples)
raw_first, raw_mid, raw_last	The individual derivative samples behind the variances
var_per_param	Variance of the derivative with respect to every parameter, in circuit order
var_mean_over_params, var_median_over_params	Mean and median of var_per_param
grad_mean_abs, cost_mean, cost_var	Mean absolute derivative, mean and variance of the cost
entropy_mean, entropy_std	Half-chain von Neumann entropy of U(θ)|0⟩ in nats (E1, E4)
haar_two_sided	Closed-form variance for a unitary 2-design on both sides of the probe (Eq. 2 of the paper)
ham, cost, variant	Cost operator, global or local cost (E2), initialisation strategy (E3)
seconds	Wall-clock time of the configuration on one CPU core

E5 training runs. cost is global or local; shots = 0 means exact gradients, otherwise Gaussian noise of standard deviation noise_sigma is added to every cost and gradient entry. grad_norm_init is the exact gradient norm at the random start, fidelity_final is |⟨0|ψ⟩|² after steps Adam steps at learning rate lr, and history is the cost after every step.

E6. var_gamma1 and var_beta1 are variances of the derivative of the cut operator C with respect to the first QAOA angles γ₁ and β₁, and raw_gamma1, raw_beta1 are the samples. The paper's figure divides them by |E|² = (3n/2)². A new random graph is drawn for every sample. In e6_qaoa_training, edges is the graph, max_cut its exact maximum cut, and the ratios are ⟨C⟩/C_max.

E7. family is hea_full {Xᵢ, Yᵢ, ZᵢZᵢ₊₁}, hea_real {Yᵢ, ZᵢZᵢ₊₁}, hva_tfim {ΣXᵢ, ΣZᵢZᵢ₊₁} or qaoa (random 3-regular graph, edges). dim is the exact DLA dimension, capped marks closures stopped at the size limit, purity_rho and purity_H are the 𝔤-purities, predicted_var is P𝔤(ρ)P𝔤(H)/dim 𝔤 (Ragone et al.), and measured_var is the cost variance over samples uniformly random parameters of an L-layer circuit.

E0. Columns {two_sided,left_only,right_only}_{measured,predicted} for hamiltonian Z⊗n (globalZ) or |0⟩⟨0|⊗n (projector), with G = Y on qubit 0.

Every table has a source_file column that names its file in raw/.

Conventions

Qubit 0 is the most significant bit, and rotations are exp(−i·θ·G/2), as in PennyLane and Qiskit. All seeds are fixed in the experiment scripts, so make results in the code repository reproduces this data (about one CPU hour).

Citation

@techreport{chongmankhong2026barren,
  author = {Chongmankhong, Thanatchon},
  title  = {Barren Plateaus from Scratch: A NumPy State-Vector Study of Trainability in Variational Quantum Circuits},
  year   = {2026},
  type   = {Technical report},
  url    = {https://thanatchon36.github.io/bp-from-scratch/paper/}
}

@software{chongmankhong2026bpfromscratch,
  author  = {Chongmankhong, Thanatchon},
  title   = {bp-from-scratch: a NumPy state-vector simulator and a reproducible study of barren plateaus},
  year    = {2026},
  version = {0.2.0},
  doi     = {10.5281/zenodo.23995487},
  url     = {https://doi.org/10.5281/zenodo.23995487}
}

License: MIT, the same as the code.