I tiakina i:
| Ngā kaituhi matua: | , , |
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| Hōputu: | Preprint |
| I whakaputaina: |
2023
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| Ngā marau: | |
| Urunga tuihono: | https://arxiv.org/abs/2303.09430 |
| Ngā Tūtohu: |
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| _version_ | 1866912904567062528 |
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| author | García-Molina, Paula Tagliacozzo, Luca García-Ripoll, Juan José |
| author_facet | García-Molina, Paula Tagliacozzo, Luca García-Ripoll, Juan José |
| contents | This work presents a comparative study of new and existing optimization and diagonalization methods for solving time-independent partial differential equations (PDEs) using matrix product states (MPS) in the quantized tensor-train formalism (QTT). This study focuses on Hamiltonian equations, for which five algorithms are introduced: explicit imaginary-time evolution methods, steepest gradient descent in conventional and optimized forms, a power method, and an explicitly restarted Arnoldi method. The first five methods are engineered using a framework of limited-precision linear algebra, in which operators -- i.e., the equation itself -- and vectors are represented using matrix product operator (MPO) and matrix product state (MPS) formalisms, and where operator-vector multiplication and vector addition are approximated with limited resources. All methods are benchmarked using an exactly solvable PDE for a quantum harmonic oscillator in one and two dimensions over a regular grid with up to $2^{30}$ points and compared with the density matrix renormalization group (DMRG) method. Our study reveals that all MPS-based techniques exponentially outperform exact diagonalization techniques based on vectors regarding memory usage. Imaginary-time algorithms are shown to underperform any gradient descent in terms of calibration needs and costs. Finally, MPS DMRG and interpolated Arnoldi-like asymptotically outperform all other methods, including state-of-the-art vector-based exact diagonalization, with significant advantages in time and memory use. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_09430 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Comparative study of matrix product state/quantized tensor-train algorithms for solving time-independent partial differential equations García-Molina, Paula Tagliacozzo, Luca García-Ripoll, Juan José Quantum Physics This work presents a comparative study of new and existing optimization and diagonalization methods for solving time-independent partial differential equations (PDEs) using matrix product states (MPS) in the quantized tensor-train formalism (QTT). This study focuses on Hamiltonian equations, for which five algorithms are introduced: explicit imaginary-time evolution methods, steepest gradient descent in conventional and optimized forms, a power method, and an explicitly restarted Arnoldi method. The first five methods are engineered using a framework of limited-precision linear algebra, in which operators -- i.e., the equation itself -- and vectors are represented using matrix product operator (MPO) and matrix product state (MPS) formalisms, and where operator-vector multiplication and vector addition are approximated with limited resources. All methods are benchmarked using an exactly solvable PDE for a quantum harmonic oscillator in one and two dimensions over a regular grid with up to $2^{30}$ points and compared with the density matrix renormalization group (DMRG) method. Our study reveals that all MPS-based techniques exponentially outperform exact diagonalization techniques based on vectors regarding memory usage. Imaginary-time algorithms are shown to underperform any gradient descent in terms of calibration needs and costs. Finally, MPS DMRG and interpolated Arnoldi-like asymptotically outperform all other methods, including state-of-the-art vector-based exact diagonalization, with significant advantages in time and memory use. |
| title | Comparative study of matrix product state/quantized tensor-train algorithms for solving time-independent partial differential equations |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2303.09430 |