Learning quantum Hamiltonians at any temperature in polynomial time with Chebyshev and bit complexity
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| Format: | Preprint |
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2024
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| _version_ | 1866910322687737856 |
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| author | Wodecki, Ales Marecek, Jakub |
| author_facet | Wodecki, Ales Marecek, Jakub |
| contents | We consider the problem of learning local quantum Hamiltonians given copies of their Gibbs state at a known inverse temperature, following Haah et al. [2108.04842] and Bakshi et al. [arXiv:2310.02243]. Our main technical contribution is a new flat polynomial approximation of the exponential function based on the Chebyshev expansion, which enables the formulation of learning quantum Hamiltonians as a polynomial optimization problem. This, in turn, can benefit from the use of moment/SOS relaxations, whose polynomial bit complexity requires careful analysis [O'Donnell, ITCS 2017]. Finally, we show that learning a $k$-local Hamiltonian, whose dual interaction graph is of bounded degree, runs in polynomial time under mild assumptions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_05552 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Learning quantum Hamiltonians at any temperature in polynomial time with Chebyshev and bit complexity Wodecki, Ales Marecek, Jakub Quantum Physics Machine Learning Optimization and Control We consider the problem of learning local quantum Hamiltonians given copies of their Gibbs state at a known inverse temperature, following Haah et al. [2108.04842] and Bakshi et al. [arXiv:2310.02243]. Our main technical contribution is a new flat polynomial approximation of the exponential function based on the Chebyshev expansion, which enables the formulation of learning quantum Hamiltonians as a polynomial optimization problem. This, in turn, can benefit from the use of moment/SOS relaxations, whose polynomial bit complexity requires careful analysis [O'Donnell, ITCS 2017]. Finally, we show that learning a $k$-local Hamiltonian, whose dual interaction graph is of bounded degree, runs in polynomial time under mild assumptions. |
| title | Learning quantum Hamiltonians at any temperature in polynomial time with Chebyshev and bit complexity |
| topic | Quantum Physics Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2402.05552 |