Hybrid quantum-classical optimization and learning strategies are among the most promising approaches to harnessing quantum information or gaining a quantum advantage over classical methods. However, efficient estimation of the gradient of the objective function in such models remains a challenge due to several factors including the exponential dimensionality of the Hilbert spaces, and information loss of quantum measurements. In this work, we study generic parameterized circuits in the context of variational methods. We develop a framework for gradient estimation that exploits the algebraic symmetries of Hamiltonian characterized through Lie algebra or group theory. Particularly, we prove that when the dimension of the dynamical Lie algebra is polynomial in the number of qubits, one can estimate the gradient with polynomial classical and quantum resources. This is done by a series of Hadamard tests applied to the output of the ansatz with no change to its circuit. We show that this approach can be equipped with classical shadow tomography to further reduce the measurement shot complexity to scale logarithmically with the number of parameters.

翻译：混合量子-经典优化与学习策略是利用量子信息或获得量子优势最有前景的方法之一。然而，由于希尔伯特空间的指数维度和量子测量的信息损失等因素，这类模型中目标函数梯度的有效估计仍是一个挑战。本文针对变分方法中的通用参数化电路展开研究，我们构建了一个梯度估计框架，该框架利用通过李代数或群论表征的哈密顿量的代数对称性。特别地，我们证明：当动态李代数的维度关于量子比特数量呈多项式增长时，可通过多项式经典与量子资源估计梯度。这一目标通过一系列对ansatz输出进行Hadamard测试实现，且无需改变其电路结构。研究表明，该方法可与经典阴影层析成像相结合，将测量样本复杂度进一步降低至与参数数量呈对数规模。