矩阵问题的量子时空权衡
Quantum Time-Space Tradeoffs for Matrix Problems
AI总结：
本文研究了矩阵问题的量子时空权衡，证明现有经典线性代数问题的紧时空下界同样适用于量子算法，说明量子计算无渐近优势，并改进了布尔矩阵乘法的下界。
摘要：
我们考虑了量子计算机解决各种涉及矩阵的问题所需的时间和空间，其中许多问题在先前的工作中仅在经典情况下进行了分析。我们的主要结果表明，对于一系列线性代数问题——包括矩阵-向量乘法、矩阵求逆、矩阵乘法和矩阵幂运算——现有的经典时空权衡（其中几个对于每个空间界限都是紧的）同样适用于量子算法。例如，对于几乎所有矩阵 $A$，包括离散傅里叶变换（DFT）矩阵，我们证明具有至多 $T$ 个输入查询和 $S$ 个量子比特内存的量子电路需要 $T=Ω(n^2/S)$ 来计算 $x \in \{0,1\}^n$ 的矩阵-向量乘积 $Ax$。我们类似地证明，对于 $n\times n$ 二进制矩阵的矩阵乘法需要 $T=Ω(n^3 / \sqrt{S})$。由于我们的许多下界与具有相同时间和空间复杂度的确定性算法相匹配，我们证明了量子计算机在任何空间界限下都无法为这些问题提供任何渐近优势。我们针对更强的量子累积内存复杂度概念——即电路每层空间的总和——获得了匹配的下界。\n我们还考虑了布尔（即 AND-OR）矩阵乘法和矩阵-向量乘积，将先前 $n\times n$ 布尔矩阵乘法的量子时空权衡下界从 $T=Ω(n^{2.5}/S^{1/2})$ 改进为 $T=Ω(n^{2.5}/S^{1/4})$。\n我们对布尔矩阵乘法改进的下界基于一种新的着色论证，该论证从先前工作中使用的强直积定理中提取了更多信息。我们对线性代数问题的紧下界需要在 Zhandry 的记录查询技术中添加一种新的分桶方法，使我们能够应用经典论证来限制量子电路的成功概率上限。
英文摘要：
We consider the time and space required for quantum computers to solve a wide variety of problems involving matrices, many of which have only been analyzed classically in prior work. Our main results show that for a range of linear algebra problems -- including matrix-vector product, matrix inversion, matrix multiplication and powering -- existing classical time-space tradeoffs, several of which are tight for every space bound, also apply to quantum algorithms. For example, for almost all matrices $A$, including the discrete Fourier transform (DFT) matrix, we prove that quantum circuits with at most $T$ input queries and $S$ qubits of memory require $T=Ω(n^2/S)$ to compute matrix-vector product $Ax$ for $x \in \{0,1\}^n$. We similarly prove that matrix multiplication for $n\times n$ binary matrices requires $T=Ω(n^3 / \sqrt{S})$. Because many of our lower bounds match deterministic algorithms with the same time and space complexity, we show that quantum computers cannot provide any asymptotic advantage for these problems with any space bound. We obtain matching lower bounds for the stronger notion of quantum cumulative memory complexity -- the sum of the space per layer of a circuit. We also consider Boolean (i.e. AND-OR) matrix multiplication and matrix-vector products, improving the previous quantum time-space tradeoff lower bounds for $n\times n$ Boolean matrix multiplication to $T=Ω(n^{2.5}/S^{1/4})$ from $T=Ω(n^{2.5}/S^{1/2})$. Our improved lower bound for Boolean matrix multiplication is based on a new coloring argument that extracts more from the strong direct product theorem used in prior work. Our tight lower bounds for linear algebra problems require adding a new bucketing method to the recording-query technique of Zhandry that lets us apply classical arguments to upper bound the success probability of quantum circuits.
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