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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.01762 |
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| _version_ | 1866929449279160320 |
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| author | Dong, Xiaojing Peng, Yizhe Tang, Qili Yang, Yin Yu, Yue |
| author_facet | Dong, Xiaojing Peng, Yizhe Tang, Qili Yang, Yin Yu, Yue |
| contents | Ref.[BCOW17] introduced a pioneering quantum approach (coined BCOW algorithm) for solving linear differential equations with optimal error tolerance. Originally designed for a specific class of diagonalizable linear differential equations, the algorithm was extended by Krovi in [Kro23] to encompass broader classes, including non-diagonalizable and even singular matrices. Despite the common misconception, the original algorithm is indeed applicable to non-diagonalizable matrices, with diagonalisation primarily serving for theoretical analyses to establish bounds on condition number and solution error. By leveraging basic estimates from [Kro23], we derive bounds comparable to those outlined in the Krovi algorithm, thereby reinstating the advantages of the BCOW approach. Furthermore, we extend the BCOW algorithm to address time-dependent linear differential equations by transforming non-autonomous systems into higher-dimensional autonomous ones, a technique also applicable for the Krovi algorithm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_01762 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Investigation on a quantum algorithm for linear differential equations Dong, Xiaojing Peng, Yizhe Tang, Qili Yang, Yin Yu, Yue Quantum Physics Ref.[BCOW17] introduced a pioneering quantum approach (coined BCOW algorithm) for solving linear differential equations with optimal error tolerance. Originally designed for a specific class of diagonalizable linear differential equations, the algorithm was extended by Krovi in [Kro23] to encompass broader classes, including non-diagonalizable and even singular matrices. Despite the common misconception, the original algorithm is indeed applicable to non-diagonalizable matrices, with diagonalisation primarily serving for theoretical analyses to establish bounds on condition number and solution error. By leveraging basic estimates from [Kro23], we derive bounds comparable to those outlined in the Krovi algorithm, thereby reinstating the advantages of the BCOW approach. Furthermore, we extend the BCOW algorithm to address time-dependent linear differential equations by transforming non-autonomous systems into higher-dimensional autonomous ones, a technique also applicable for the Krovi algorithm. |
| title | Investigation on a quantum algorithm for linear differential equations |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2408.01762 |