Add Figure 2 QNTK diagnostics reproduction

[ninjaduck7]

Summary

• add a self-contained TensorCircuit-NG reproduction of Figure 2 from
  "Towards Practical Quantum Neural Network Diagnostics with Neural Tangent
  Kernels" (arXiv:2503.01966)
• include the four QNTK diagnostics for low/high-frequency HVA and HEA circuits
• include gallery metadata and the generated Figure 2 output

Reproduction scope

The example uses the paper's 6-qubit transverse-field Ising regression task,
four QNN architectures, depths 5 through 30, and three fixed random
initializations. This reduced scope covers the portion visible on the paper's
approximately 620-parameter axis.

The implementation uses TensorCircuit-NG circuits, expectations, and PyTorch
automatic differentiation. It runs on CUDA when available and falls back to
CPU.

Results

The reproduction recovers the main Figure 2 behavior:

• low-frequency circuits retain high test R2
• high-frequency circuits approach test R2 near zero as depth increases
• QNTK spectral and conditioning diagnostics separate the two regimes

Validation

• all 72 configured reproduction points completed successfully
• generated Figure 2 output is included in outputs/result.png
• gallery metadata and generated gallery assets are included
• Black check passed
• Pylint passed with 10.00/10
• Python compilation passed
• gallery generation passed

Reproduction command

python examples/reproduce_papers/2025_qntk_diagnostics/main.py

Notes

The manuscript does not publish the VQE target table, initialization details,
observable choice, or an explicit feature-to-angle conversion. The example
documents these reconstruction choices and uses digitized VQE targets from the
paper’s vector dataset figure.
Closes #98

[refraction-ray]

Thanks for the contribution, the following are some suggestions

Critical: wrong expectation observable

examples/reproduce_papers/2025_qntk_diagnostics/main.py:147-159 evaluates expectation_ps(z=[0]), while the paper’s TFIM target is the
transverse magnetization (\langle\sigma_X\rangle).

This means the script computes the QNTK of a different model.

Performance/reproducibility suggestion: make the script backend agnostic

The script currently hard-codes PyTorch through torch.func.jacrev, torch.as_tensor, device setup, and tensor conversion. Move these
operations to tc.backend:

• tc.backend.convert_to_tensor
• tc.backend.jacrev
• tc.backend.vmap
• tc.backend.jit

Then expose a backend option for identical JAX/PyTorch runs. The benchmark should report JAX compilation time separately from steady-
state execution time and use identical precision, seeds, circuit depths, and encoding scale. The current Python loop in batch_outputs
should be replaced by backend vmap.

[ninjaduck7]

Thanks for the detailed review. I have pushed an update in commit 272271f.

• Corrected the observable from expectation_ps(z=[0]) to
  expectation_ps(x=[0]), matching the transverse-X target.
• Replaced direct PyTorch tensor/device handling with TensorCircuit backend
  APIs, including convert_to_tensor, vmap, and jit.
• Added a --backend jax|pytorch option.
• Kept the same precision, seeds, depths, inputs, and encoding scale for
  comparable backend runs.

JAX uses tc.backend.jacrev. For reproducibility, the tests used
TensorCircuit-NG 1.9.1 and PyTorch 2.7.1+cu128. The PyTorch
tc.backend.jacrev failure is not CUDA-specific: the same minimal test
fails on both CPU and CUDA. In TCNG 1.9.1, jacrev combines vmap with a
PyTorch backend vjp implemented through torch.autograd.functional.vjp,
which is incompatible with functorch transforms. Therefore, the PyTorch
path uses the equivalent tc.backend.grad transform for this scalar-output
Jacobian.

Validation completed:

• PyTorch backend: 72/72 configured points completed successfully.
• Representative JAX/PyTorch parity checks covered all four architectures.
• Maximum output and Jacobian differences were below 5e-16.
• The benchmark uses identical precision, seed, depths, and encoding scale,
  and reports JAX compilation separately from steady-state execution.
• No TensorCircuit-NG core source files were modified.

The scalar observable is currently the representative first-qubit
transverse-X expectation (X_0). If the intended observable is the
normalized six-qubit average of all X expectations, please let me know.