Efficient Quantum Gradient and Higher-Order Derivative Estimation Via Generalized Hadamard Test

Abstract

In the context of Noisy Intermediate-Scale Quantum (NISQ) computing, parameterized quantum circuits (PQCs) are a promising paradigm for quantum sensing, optimal control, optimization, and machine learning on near-term quantum hardware. Gradient-based methods are crucial for analyzing PQCs and offer significant advantages in the convergence rates of Variational Quantum Algorithms (VQAs) compared to gradient-free methods. However, existing gradient estimation techniques-such as Finite Difference, Parameter Shift Rule, Hadamard Test, and Direct Hadamard Test-can yield suboptimal circuits for certain PQCs. To address these limitations, we introduce the Flexible Hadamard Test, which can invert the roles of ansatz generators and observables in first-order gradient estimation, enabling the use of measurement optimization techniques to improve circuit efficiency. For higher-order derivatives, we propose the $k$-fold Hadamard Test, which evaluates higher-order partial derivatives using a single quantum circuit. Additionally, we present Quantum Automatic Differentiation (QAD)-a unified gradient estimation framework that adaptively selects the most efficient method for each parameter in a PQC, departing from the standard practice of using a single method uniformly. Through extensive numerical experiments, we demonstrate that our proposed methods yield up to a linear-factor reduction in circuit execution cost for real VQA applications. These advances accelerate quantum algorithm training and provide practical utility in the NISQ era.