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.

学习局部量子哈密顿量的问题，基于已知逆温度下的吉布斯态，通过使用 Chebyshev 展开的新型平面多项式逼近指数函数，我们将学习量子哈密顿量的表述转化为多项式优化问题，然后利用时刻 / 二次松弛方法加快求解速度，对于度受限的双重交互图，我们证明在温和假设下学习 $k$ 局部哈密顿量可以在多项式时间内完成。

利用 Chebyshev 和位复杂度在多项式时间内学习任意温度下的量子哈密顿量

Learning quantum Hamiltonians at any temperature in polynomial time with  Chebyshev and bit complexity

Formulating the alternating current optimal power flow (ACOPF) as a
polynomial optimization problem makes it possible to solve large instances in
practice and to guarantee asymptotic convergence in theory.

用多项式优化问题表达交流最优电力流（ACOPF）问题可以在实践中解决大规模实例，并在理论上保证渐近收敛。

多项式优化问题中的最优潮流

Optimal Power Flow as a Polynomial Optimization Problem

Consider the polynomial optimization problem whose objective and constraints
are all described by multivariate polynomials. Under some genericity
assumptions, %% on these polynomials, we prove that the optimality conditions
always hold on optimizers, and the coordinates of optimizers are algebraic
functions of the coefficients of the input polynomials. We also give a general
formula for the algebraic degree of the optimal coordinates. The derivation of
the algebraic degree is equivalent to counting the number of all complex
critical points. As special cases, we obtain the algebraic degrees of
quadratically constrained quadratic programming (QCQP), second order cone
programming (SOCP) and $p$-th order cone programming (pOCP), in analogy to the
algebraic degree of semidefinite programming.

针对多项式优化问题，本文证明了在一些范畴的多项式的约束下，优化条件总是满足优化器，优化器坐标是输入多项式系数的代数函数，并且给出了最优坐标的代数度数的一般公式，该公式的推导类似于计算所有复的临界点的数量。作为特例，我们还获得了类比于半定规划的代数度量的 QCQP、SOCP 和 pOCP 的代数度量。