  Variational quantum eigensolver (VQE) optimizes parameterized eigenstates of
a Hamiltonian on a quantum processor by updating parameters with a classical
computer. Such a hybrid quantum-classical optimization serves as a practical
way to leverage up classical algorithms to exploit the power of near-term
quantum computing. Here, we develop a hybrid algorithm for VQE, emphasizing the
classical side, that can solve a group of related Hamiltonians simultaneously.
The algorithm incorporates a snake algorithm into many VQE tasks to
collectively optimize variational parameters of different Hamiltonians. Such
so-called collective VQEs~(cVQEs) is applied for solving molecules with varied
bond length, which is a standard problem in quantum chemistry. Numeral
simulations show that cVQE is not only more efficient in convergence, but also
trends to avoid single VQE task to be trapped in local minimums. The collective
optimization utilizes intrinsic relations between related tasks and may inspire
advanced hybrid quantum-classical algorithms for solving practical problems.
