Learning quantum Hamiltonians at any temperature in polynomial time
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911659207950336 |
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| author | Bakshi, Ainesh Liu, Allen Moitra, Ankur Tang, Ewin |
| author_facet | Bakshi, Ainesh Liu, Allen Moitra, Ankur Tang, Ewin |
| contents | We study the problem of learning a local quantum Hamiltonian $H$ given copies of its Gibbs state $ρ= e^{-βH}/\textrm{tr}(e^{-βH})$ at a known inverse temperature $β>0$. Anshu, Arunachalam, Kuwahara, and Soleimanifar (arXiv:2004.07266) gave an algorithm to learn a Hamiltonian on $n$ qubits to precision $ε$ with only polynomially many copies of the Gibbs state, but which takes exponential time. Obtaining a computationally efficient algorithm has been a major open problem [Alhambra'22 (arXiv:2204.08349)], [Anshu, Arunachalam'22 (arXiv:2204.08349)], with prior work only resolving this in the limited cases of high temperature [Haah, Kothari, Tang'21 (arXiv:2108.04842)] or commuting terms [Anshu, Arunachalam, Kuwahara, Soleimanifar'21]. We fully resolve this problem, giving a polynomial time algorithm for learning $H$ to precision $ε$ from polynomially many copies of the Gibbs state at any constant $β> 0$.
Our main technical contribution is a new flat polynomial approximation to the exponential function, and a translation between multi-variate scalar polynomials and nested commutators. This enables us to formulate Hamiltonian learning as a polynomial system. We then show that solving a low-degree sum-of-squares relaxation of this polynomial system suffices to accurately learn the Hamiltonian. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_02243 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Learning quantum Hamiltonians at any temperature in polynomial time Bakshi, Ainesh Liu, Allen Moitra, Ankur Tang, Ewin Quantum Physics Data Structures and Algorithms Machine Learning We study the problem of learning a local quantum Hamiltonian $H$ given copies of its Gibbs state $ρ= e^{-βH}/\textrm{tr}(e^{-βH})$ at a known inverse temperature $β>0$. Anshu, Arunachalam, Kuwahara, and Soleimanifar (arXiv:2004.07266) gave an algorithm to learn a Hamiltonian on $n$ qubits to precision $ε$ with only polynomially many copies of the Gibbs state, but which takes exponential time. Obtaining a computationally efficient algorithm has been a major open problem [Alhambra'22 (arXiv:2204.08349)], [Anshu, Arunachalam'22 (arXiv:2204.08349)], with prior work only resolving this in the limited cases of high temperature [Haah, Kothari, Tang'21 (arXiv:2108.04842)] or commuting terms [Anshu, Arunachalam, Kuwahara, Soleimanifar'21]. We fully resolve this problem, giving a polynomial time algorithm for learning $H$ to precision $ε$ from polynomially many copies of the Gibbs state at any constant $β> 0$. Our main technical contribution is a new flat polynomial approximation to the exponential function, and a translation between multi-variate scalar polynomials and nested commutators. This enables us to formulate Hamiltonian learning as a polynomial system. We then show that solving a low-degree sum-of-squares relaxation of this polynomial system suffices to accurately learn the Hamiltonian. |
| title | Learning quantum Hamiltonians at any temperature in polynomial time |
| topic | Quantum Physics Data Structures and Algorithms Machine Learning |
| url | https://arxiv.org/abs/2310.02243 |