Localizing SLD Geometry for Scalable Quantum Natural Gradient under Circuit Noise

Variational quantum circuits generally produce mixed states in the presence of circuit noise. Quantum natural gradient (QNG) accounts for this mixed-state geometry using the symmetric-logarithmic-derivative (SLD) quantum Fisher information matrix, whose direct estimation can exceed practical circuit-shot budgets. Tractable QNG approximations developed for pure states do not directly provide this geometry, while mixed-state approaches often use more accessible surrogates, such as the Hilbert–Schmidt metric. We propose an approach to approximate the SLD geometry using local reductions of the noisy state. We define graph-local physical windows of radius $r$ and restrict the local SLD score space within each window to centered Pauli strings of weight at most $k_S$. We assemble the local SLD metrics across fixed parameter blocks to obtain a block-local approximation. Increasing $r$ or $k_S$ yields nested approximation hierarchies, and the complete centered Pauli score space recovers the exact reduced-state SLD metric when no score-space regularization is used. For finite-shot implementation, we estimate the score-Gram and response matrices using local Pauli and parameter-shift measurements without reconstructing the reduced states. We derive finite-shot metric-estimation error bounds and show that, under fixed-locality assumptions, a sufficient metric-estimation cost scales as $O(n\log n)$ for fixed accuracy and confidence. We also establish convergence of the regularized finite-shot update to a stationary neighborhood of the noisy objective under standard stochastic-optimization assumptions. Experiments on noisy variational quantum eigensolver (VQE) instances for the Heisenberg model show improved convergence per cumulative shot compared with SGD, Adam, and approximate mixed-state QNG baselines under matched finite-shot budgets.