arXiv:2310.02243quant-ph
Learning quantum Hamiltonians at any temperature in polynomial time
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Abstract
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.
Parameter estimation algorithms: "Learning quantum Hamiltonians at any temperature in polynomial time" (2024) https://arxiv.org/abs/2310.02243 .. https://news.ycombinator.com/item?id=40396171 "Robustly learning Hamiltonian dynamics of a superconducting quantum processor" (2024)…
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