Efficient Quantum Gradient and Higher-order Derivative Estimation via Generalized Hadamard Test

In Noisy Intermediate-Scale Quantum (NISQ) computing, parameterized quantum circuits (PQCs) are a promising approach for applications in quantum sensing, control, optimization, and machine learning. Current gradient estimation methods, including Finite Difference and Parameter Shift Rule, are often inefficient for certain PQCs. To address this, we introduce the Flexible Hadamard Test, which inverts roles between ansatz generators and observables, enabling optimized measurement techniques for efficient gradient computation. For higher-order derivatives, we propose the k-fold Hadamard Test, achieving efficient k-order derivatives with a single circuit. Additionally, we develop Quantum Automatic Differentiation (QAD), the first adaptive method to choose the optimal gradient estimation technique per parameter. Numerical results show our approach can reduce circuit executions by up to an O(N) factor, accelerating Variational Quantum Algorithm performance in NISQ settings.