Gradient Descent (VQE)

Layered $R_Y$-$CX$ ansatz minimising $⟨Z\otimes I\otimes \cdots ⟩$. Full optimisation loop timed end-to-end.

qudit and PennyLane use PyTorch autograd — one backward pass per step. Others use parameter-shift — $2L$ forward passes per step where $L$ = number of parameters.

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Notes:

• qudit (AD) — torch.autograd through nn.Module. One backward pass per step regardless of parameter count.
• pennylane (AD) — same gradient method, higher constant overhead from the QNode abstraction.
• qiskit (PS) — StatevectorEstimator batch API with parameter-shift. Batches all shift evaluations per step.
• cirq (PS), braket (PS), qutip (PS) — parameter-shift with sequential circuit rebuilds per evaluation. Cost is $2L\times$ forward cost per step.