  We propose a nonvariational scheme for geometry optimization of molecules for
the first-quantized eigensolver, a recently proposed framework for quantum
chemistry using the probabilistic imaginary-time evolution (PITE) on a quantum
computer. While the electrons in a molecule are treated in the scheme as
quantum mechanical particles, the nuclei are treated as classical point
charges. We encode both electronic states and candidate molecular geometries as
a superposition of many-qubit states, leading to quantum advantage. The
histogram formed by outcomes of repeated measurements gives the global minimum
of the energy surface. We demonstrate that the circuit depth scales as O (n_e^2
poly(log n_e)) for the electron number n_e, which can be reduced to O (n_e
poly(log n_e)) if extra O (n_e log n_e) qubits are available. We corroborate
the scheme via numerical simulations. The new efficient scheme will be helpful
for achieving scalability of practical quantum chemistry on quantum computers.
As a special case of the scheme, a classical system composed only of charged
particles is admitted. We also examine the scheme adapted to variational
calculations that prioritize saving circuit depths for noisy intermediate-scale
quantum (NISQ) devices.
