经典多项式优化的量子估计
Quantum estimates for classical polynomial optimization
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中文总结 AI 辅助
研究多元齐次多项式上下界问题，引入受量子力学变分方法启发的策略，通过将多项式替换为量子算子并对角化，近似系数张量特征值来求界，成功应用于标准测试及数学物理相关问题。
中文摘要 AI 辅助
寻找多元齐次多项式的上下界问题既困难又重要，其应用广泛。从张量特征值理论看，该问题等同于找到给定多项式对应系数张量的最小和最大特征值。文献中的标准方法是进行非线性迭代以寻找多项式增长最快或最慢的最优射线，但高阶张量算法的收敛性不稳定。本文引入受量子力学变分方法启发的不同策略，将原多项式替换为作用在合适大状态空间的算子，通过对该量子算子对角化来近似系数张量的特征值，从而找到多项式的界，并成功应用于标准测试例子及数学物理中的其他问题。
英文摘要
The problem of finding lower and upper bounds on multivariate homogeneous polynomials is both difficult and important given its applications to questions ranging from dynamical stability in complex potential landscapes to data analysis. From the standpoint of tensor eigenvalue theory, the question is equivalent to finding the smallest and the largest eigenvalues of the coefficient tensor corresponding to the given polynomial. Standard approaches outlined in the literature amount to running nonlinear iterations in search for the optimal rays along which the growth of the polynomial is fastest or slowest. Unlike the case of matrices (or their corresponding multivariate quadratic forms) convergence of such algorithms for higher-rank tensors is capricious due to the complex topography of polynomial objective functions. In this essay, a very different strategy, inspired by quantum-mechanical variational methods, is introduced for finding bounds on polynomials. The original polynomial is replaced by an operator acting in a suitably chosen (large) space of states, such that in an appropriate "classical" limit this operator approaches the original polynomial expression made of commutative variables. As a result, approximating the smallest and largest eigenvalues of the coefficient tensor, and thus finding bounds on polynomials, amounts to diagonalizing the resulting quantum operator, represented as a large matrix, and then inspecting the smallest and largest eigenvalues of this matrix. This approach is then successfully applied to standard test examples from tensor eigenvalue literature and other problems of interest in mathematical physics including Strichartz-type inequalities.
