Exhaustive search for optimal molecular geometries using imaginary-time evolution on a quantum computer

Abstract

Abstract This study proposes a nonvariational scheme for geometry optimization of molecules for the first-quantized eigensolver, which is a recently proposed framework for quantum chemistry using probabilistic imaginary-time evolution (PITE). In this scheme, the nuclei in a molecule are treated as classical point charges while the electrons are treated as quantum mechanical particles. The electronic states and candidate geometries are encoded as a superposition of many-qubit states, for which a histogram created from repeated measurements gives the global minimum of the energy surface. We demonstrate that the circuit depth per step scales as $${{{\mathcal{O}}}}({n}_{{\rm {e}}}^{2}{{{\rm{poly}}}}(\log {n}_{{\rm {e}}}))$$ O ( n e 2 poly ( log n e ) ) for the electron number n e, which can be reduced to $${{{\mathcal{O}}}}({n}_{{\rm {e}}}{{{\rm{poly}}}}(\log {n}_{{\rm {e}}}))$$ O ( n e poly ( log n e ) ) if extra $${{{\mathcal{O}}}}({n}_{{\rm {e}}}\log {n}_{{\rm {e}}})$$ O ( n e log n e ) qubits are available. Moreover, resource estimation implies that the total computational time of our scheme starting from a good initial guess may exhibit overall quantum advantage in molecule size and candidate number. The proposed scheme is corroborated using numerical simulations. Additionally, a scheme adapted to variational calculations is examined that prioritizes saving circuit depths for noisy intermediate-scale quantum (NISQ) devices. A classical system composed only of charged particles is considered as a special case of the scheme. The new efficient scheme will assist in achieving scalability in practical quantum chemistry on quantum computers.