Numerical investigation of the quantum inverse algorithm on small molecules

We evaluate the accuracy of the quantum inverse (Q-Inv) algorithm in which the multiplication of $\hat{H}^{-k}$ to the reference wavefunction is replaced by the Fourier Transformed multiplication of $e^{-i\lambda \hat{H}}$, as a function of the integration parameters ($\lambda$) and the power $k$ for various systems, including H$_2$, LiH, BeH$_2$ and the notorious H$_4$ molecule at single point. We further consider the possibility of employing the Gaussian-quadrature rule as an alternate integration method and compared it to the results employing trapezoidal integration. The Q-Inv algorithm is compared to the inverse iteration method using the $\hat{H}^{-1}$ inverse (I-Iter) and the exact inverse by lower-upper decomposition (LU). Energy values are evaluated as the expectation values of the Hamiltonian. Results suggest that the Q-Inv method provides lower energy results than the I-Iter method up to a certain $k$, after which the energy increases due to errors in the numerical integration that are dependent of the integration interval. A combined Gaussian-quadrature and trapezoidal integration method proved to be more effective at reaching convergence while decreasing the number of operations. For systems like H$_4$, in which the Q-Inv can not reach the expected error threshold, we propose a combination of Q-Inv and I-Iter methods to further decrease the error with $k$ at lower computational cost. Finally, we summarize the recommended procedure when treating unknown systems.