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...

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Bibliographic Details
Published in:arXiv.org 2024-04
Main Authors: Cainelli, Mauro, Baba, Reo, Kurashige, Yuki
Format: Article
Language:English
Subjects:
Algorithms
Energy value
Hamiltonian functions
Numerical integration
Quadratures
System effectiveness
Wave functions
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Summary: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.
ISSN:2331-8422