Gradient estimation is a central challenge in training parameterized quantum circuits (PQCs) for hybrid quantum-classical optimization and learning problems. This difficulty arises from several factors, including the exponential dimensionality of the Hilbert spaces and the information loss in quantum measurements. Existing estimators, such as finite difference and the parameter shift rule, often fail to adequately address these challenges for certain classes of PQCs. In this work, we propose a novel gradient estimation framework that leverages the underlying Lie algebraic structure of PQCs, combined with the Hadamard test. By analyzing the differential of the matrix exponential, we derive an expression for the gradient as a linear combination of expectation values obtained via Hadamard tests. The coefficients in this decomposition depend solely on the circuit's parameterization and can be estimated using state-of-the-art shadow tomography techniques. Hence, our approach enables efficient gradient estimation, requiring a number of measurement shots that scales logarithmically with the number of parameters, and with polynomial classical and quantum time. This is an exponential reduction in the measurement cost and a polynomial speed-up in time compared to existing works.

翻译：梯度估计是训练参数化量子电路用于混合量子-经典优化与学习问题的核心挑战。这一困难源于多重因素，包括希尔伯特空间的指数维度以及量子测量中的信息损失。现有的估计方法，如有限差分法和参数位移法则，往往无法充分应对某些类型参数化量子电路所面临的挑战。本文提出了一种新型梯度估计框架，该框架利用了参数化量子电路底层的李代数结构，并结合了Hadamard测试。通过分析矩阵指数的微分，我们将梯度推导为通过Hadamard测试获得的期望值的线性组合。此分解中的系数仅取决于电路的参数化方式，并可通过最先进的影子断层扫描技术进行估计。因此，我们的方法实现了高效的梯度估计，所需的测量次数随参数数量呈对数增长，且经典与量子时间开销均为多项式级别。相较于现有工作，这实现了测量成本的指数级降低和时间的多项式级加速。