凸优化与实矩阵-向量查询问题的量子下界
Quantum lower bounds for convex optimization and real matrix-vector query problems
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中文总结 AI 辅助
本文提出行列式见证方法，证明凸函数最小化及实矩阵求逆等问题的量子查询复杂度下界为$\tilde\u03a9(n)$，并给出迹、行列式符号及大小的近线性下界。
中文摘要 AI 辅助
我们（作者及承担繁重工作的AI系统）证明，在$\u211d^n$的凸子集上，使用评估查询和成员资格查询最小化凸函数的量子查询复杂度为$\tilde\u03a9(n)$，几乎匹配已知的最佳上界。特别地，我们证明即使对于二次最小化（这等价于使用矩阵-向量查询求逆一个$n \times n$实矩阵）也是如此。我们还证明了在矩阵-向量查询模型中，计算实矩阵的迹、行列式的符号以及行列式的大小的量子查询复杂度具有线性或近线性下界。我们使用一种新颖的量子下界技术——行列式见证方法，该方法基于识别一个见证，其傅里叶变换在低秩矩阵上消失，并且与所计算的函数相关性良好。
英文摘要
We (the author and the AI systems that did the heavy lifting) show that the quantum query complexity of minimizing a convex function over a convex subset of $\mathbb{R}^n$ with evaluation and membership queries is $\tildeΩ(n)$, nearly matching the best known upper bound. In particular, we show this even for quadratic minimization, which is equivalent to inverting an $n \times n$ real matrix using matrix-vector queries. We also show linear or nearly linear lower bounds on the quantum query complexity of computing the trace, the sign of the determinant, and the magnitude of the determinant of a real matrix in the matrix-vector query model. We use a novel quantum lower bound technique, the determinantal witness method, based on identifying a witness whose Fourier transform vanishes on low-rank matrices and that correlates well with the function being computed.
发表机构
- Joint Center for Quantum Information and Computer Science（量子信息与计算科学联合中心）
- University of Maryland（马里兰大学）
机构由 AI 辅助整理，请以论文原文为准。
