  Variational quantum eigensolver~(VQE) typically optimizes variational
parameters in a quantum circuit to prepare eigenstates for a quantum system.
Its applications to many problems may involve a group of Hamiltonians, e.g.,
Hamiltonian of a molecule is a function of nuclear configurations. In this
paper, we incorporate derivatives of Hamiltonian into VQE and develop some
hybrid quantum-classical algorithms, which explores both Hamiltonian and
wavefunction spaces for optimization. Aiming for solving quantum chemistry
problems more efficiently, we first propose mutual gradient descent algorithm
for geometry optimization by updating parameters of Hamiltonian and
wavefunction alternatively, which shows a rapid convergence towards equilibrium
structures of molecules. We then establish differential equations that governs
how optimized variational parameters of wavefunction change with intrinsic
parameters of the Hamiltonian, which can speed up calculation of energy
potential surface. Our studies suggest a direction of hybrid quantum-classical
algorithm for solving quantum systems more efficiently by considering spaces of
both Hamiltonian and wavefunction.
