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 $T=\Omega(n^{2.5}/S^{1/3})$ 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问题所需的时间和空间复杂度，其中多数问题此前仅在经典计算框架下被分析。核心结果表明：对于一系列线性代数问题（包括矩阵-向量乘积、矩阵求逆、矩阵乘法与矩阵幂运算），现有的经典时间-空间权衡（其中若干结果对任意空间界均为紧界）同样适用于量子算法。例如，对几乎所有的矩阵$A$（包括离散傅里叶变换矩阵），我们证明：具有至多$T$次输入查询和$S$量子比特存储的量子电路，若计算$x \in \{0,1\}^n$的矩阵-向量乘积$Ax$，则需满足$T=\Omega(n^2/S)$。类似地，我们证明了$n\times n$二进制矩阵乘法需满足$T=\Omega(n^3 / \sqrt{S})$。由于我们的许多下界与具有相同时间-空间复杂度的确定性算法匹配，这表明量子计算机在任意空间界下均无法为这些问题提供渐进优势。对于更强的量子累积存储复杂度（即电路每层空间的累加和），我们也获得了匹配的下界。同时，研究布尔（即AND-OR）矩阵乘法与矩阵-向量乘积时，我们将$n\times n$布尔矩阵乘法的前向量子时间-空间下界从$T=\Omega(n^{2.5}/S^{1/2})$改进为$T=\Omega(n^{2.5}/S^{1/3})$。这一改进基于新的着色论证，从先前工作使用的强直积定理中提取了更多信息。针对线性代数问题的紧下界需要向Zhandry的录音查询技术引入一种新的分桶方法，使我们能够运用经典论证技巧来界定量子电路的成功概率上限。