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 $T=\Omega(n^2/S)$ 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 $T=\Omega(n^3 / \sqrt{S})$. 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 $T=\Omega(n^{2.5}/S^{1/4})$ from $T=\Omega(n^{2.5}/S^{1/2})$. 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.

翻译：我们研究了量子计算机解决多种矩阵相关问题所需的时间与空间资源，其中许多问题先前仅在经典计算框架下被分析。主要结果表明，对于一系列线性代数问题（包括矩阵-向量乘积、矩阵求逆、矩阵乘法与幂运算），现有经典时间-空间权衡（其中若干权衡在任意空间约束下均为紧界）同样适用于量子算法。例如，对于几乎所有矩阵$A$（包含离散傅里叶变换矩阵），我们证明：内存为$S$量子比特、输入查询次数至多为$T$的量子电路，要计算$x \in \{0,1\}^n$上的矩阵-向量乘积$Ax$，需满足$T=\Omega(n^2/S)$。类似地，我们证明$n\times n$二进制矩阵的乘法运算需满足$T=\Omega(n^3 / \sqrt{S})$。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，这表明量子计算机在任何空间约束下均无法为这些问题提供渐近优势。我们进一步针对更强的量子累积内存复杂度（即电路每层空间量之和）获得了匹配下界。针对布尔矩阵乘法与矩阵-向量乘积，我们将$n\times n$布尔矩阵乘法的量子时间-空间权衡下界从此前的$T=\Omega(n^{2.5}/S^{1/2})$提升至$T=\Omega(n^{2.5}/S^{1/4})$。该改进基于一种新的着色论证方法，从先前工作使用的强直积定理中提取了更多信息。为获得线性代数问题的紧下界，我们向Zhandry的记录查询技术中引入了分桶方法，从而能应用经典论证来上界量子电路的成功概率。