SGIQ: Sparse-Gradient-Informed Gradient Estimation for Variational Quantum Circuits

Gradient estimation is a major measurement bottleneck in training complex variational quantum circuits (VQCs) because backpropagation scaling is unattainable for general cases with unknown quantum inputs and no quantum memory. Existing gradient estimators therefore face a cost-accuracy trade-off: parameter-shift rule (PSR) incurs a cost linear in the parameter dimension, while simultaneous perturbation stochastic approximation (SPSA) uses few evaluations but can suffer from dimension-dependent variance. To improve this trade-off, we exploit settings in which loss-gradient energy concentrates on a small subset of parameters. We formalize this property through gradient-energy sparsity and derive sufficient conditions from causal-cone, commutation, and response-decay mechanisms. Building on this structure, we propose $\text{SGiQ}$, a $\text{S}$parse-$\text{G}$radient-$\text{i}$nformed $\text{Q}$uantum gradient estimation algorithm that uses averaged SPSA estimates to rank all circuit parameters, replaces the selected coordinate estimates with PSR gradients, and retains the SPSA estimates elsewhere. Its cost scales with the circuit's gradient structure, rather than the parameter dimension. In ideal and finite-shot regression experiments, SGiQ uses fewer gradient circuits than PSR and attains lower normalized test mean absolute error (MAE) than SPSA baselines.