多项式时间算法用于核张量范数与多体可分性

中文摘要 AI 辅助

我们研究了一个关于具有有界Frobenius范数的张量的多线性优化问题，其中每个维度为d的k个局部因子从凸集中选取。我们给出一个确定性算法，运行时间为d^{O(k)}，用于常数加性近似。作为应用，我们首次获得了在常数精度下对高阶张量的核范数单位球的弱成员判定问题以及Frobenius范数下多体量子可分性问题的多项式时间算法。我们的方法将张量优化视为一个合作式多证明者博弈，并将这一视角与递归谱压缩过程相结合。随后我们证明，当输入态通过副本而非显式经典描述给出时，相关思想可实现量子实现。在这些设置中，我们获得了一个算法，使用poly(k)个副本和poly(k log d)时间解决Frobenius范数可分性测试。

英文摘要

We study a multilinear optimization problem for tensors with bounded Frobenius norm, where each of the $k$ local factors of dimension $d$ is chosen from a convex set. We give a deterministic algorithm running in time $d^{O(k)}$ for constant additive approximations. As applications, we obtain the first polynomial-time algorithms at constant accuracy for weak membership in the nuclear-norm unit ball of high-order tensors and for multipartite quantum separability in Frobenius norm. Our approach views tensor optimization as a cooperative multiprover game and combines this perspective with a recursive spectral compression procedure. We then show that related ideas admit a quantum implementation when the input state is given through copies, rather than an explicit classical description. In these settings we obtain an algorithm to solve Frobenius-norm separability testing using $\text{poly}(k)$ copies and $\text{poly}(k\log d)$ time.

发表机构

• Bocconi University（博科尼大学）

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