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

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 $${{{\mathcal{O}}}}({n}_{{\rm {e}}}^{2}{{{\rm{poly}}}}(\log {n}_{{\rm {e}}}))$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:msubsup> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>e</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> <mml:mi>poly</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>log</mml:mi> <mml:msub> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>e</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> for the electron number n e , which can be reduced to $${{{\mathcal{O}}}}({n}_{{\rm {e}}}{{{\rm{poly}}}}(\log {n}_{{\rm {e}}}))$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>e</mml:mi> </mml:mrow> </mml:msub> <mml:mi>poly</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>log</mml:mi> <mml:msub> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>e</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> if extra $${{{\mathcal{O}}}}({n}_{{\rm {e}}}\log {n}_{{\rm {e}}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>e</mml:mi> </mml:mrow> </mml:msub> <mml:mi>log</mml:mi> <mml:msub> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>e</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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.