Numerical investigation of the quantum inverse algorithm on small molecules
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912856045256704 |
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| author | Cainelli, Mauro Baba, Reo Kurashige, Yuki |
| author_facet | Cainelli, Mauro Baba, Reo Kurashige, Yuki |
| contents | 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 $e^{-iλ\hat{H}}$, as a function of the integration parameters ($λ$) 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_07512 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Numerical investigation of the quantum inverse algorithm on small molecules Cainelli, Mauro Baba, Reo Kurashige, Yuki Chemical Physics Computational Physics Quantum Physics 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 $e^{-iλ\hat{H}}$, as a function of the integration parameters ($λ$) 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. |
| title | Numerical investigation of the quantum inverse algorithm on small molecules |
| topic | Chemical Physics Computational Physics Quantum Physics |
| url | https://arxiv.org/abs/2404.07512 |