WA-QNG¶

Implements WA-QNG, a weighted approximate quantum natural gradient for the variational quantum eigensolver.

Quantum natural gradient preconditions the gradient by the inverse of the quantum Fisher information matrix \(F\) of the full state, which is expensive to estimate. WA-QNG replaces \(F\) with a cheap surrogate built from subsystem metric tensors, one per term of the decomposed Hamiltonian \(H=\sum_m h_m H_m\). Each subsystem metric \(T_m\) is weighted by \(h_m^2\), so terms with larger Hamiltonian coefficients contribute more to the geometry, and the normalization makes the surrogate reduce to standard QNG when every term acts on the full system.

\[ \begin{aligned} W &= \frac{2}{\sum_m h_m^2}\sum_m h_m^2\, T_m, \qquad (T_m)_{ij} = \mathrm{tr}\!\left(\partial_i \rho_m\, \partial_j \rho_m\right),\\ \theta_{t+1} &= \theta_t - \eta\, W^{+}\, g_t. \end{aligned} \]

where \(\theta\) are the circuit parameters, \(\eta\) is the learning rate, \(g_t=\nabla f(\theta_t)\) is the gradient of the objective, \(h_m\) are the Hamiltonian coefficients, \(\rho_m\) is the reduced density matrix of subsystem \(m\), \(T_m\) is its Hilbert-Schmidt metric tensor, and \(W^{+}\) is the Moore-Penrose pseudoinverse of the weighted metric \(W\).

Reference: Chenyu Shi, Vedran Dunjko, Hao Wang, "Weighted Approximate Quantum Natural Gradient for Variational Quantum Eigensolver", arXiv 2025. https://arxiv.org/abs/2504.04932

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