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.

翻译：我们研究在已知逆温度$β>0$条件下，根据吉布斯态$ρ= e^{-βH}/\textrm{tr}(e^{-βH})$的副本学习局部量子哈密顿量$H$的问题。Anshu、Arunachalam、Kuwahara和Soleimanifar（arXiv:2004.07266）提出了一种算法，仅需多项式数量的吉布斯态副本即可将$n$量子比特的哈密顿量学习至精度$ε$，但该算法需要指数级时间。获得计算高效的算法一直是一个重要公开问题[Alhambra'22 (arXiv:2204.08349)]，[Anshu, Arunachalam'22 (arXiv:2204.08349)]，先前的工作仅在高温[Haah, Kothari, Tang'21 (arXiv:2108.04842)]或对易项[Anshu, Arunachalam, Kuwahara, Soleimanifar'21]的有限情况下解决了这一问题。我们完全解决了该问题，给出一个多项式时间算法，从任意常数$β>0$时多项式数量的吉布斯态副本中学习$H$至精度$ε$。我们的主要技术贡献是指数函数的新平坦多项式近似，以及多元标量多项式与嵌套对易子之间的转换。这使我们能够将哈密顿量学习表述为多项式系统。随后我们证明，求解该多项式系统的低阶平方和松弛就足以精确学习哈密顿量。