  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.
