We study quantum algorithms that learn properties of a matrix using queries that return its action on an input vector. We show that for various problems, including computing the trace, determinant, or rank of a matrix or solving a linear system that it specifies, quantum computers do not provide an asymptotic speedup over classical computation. On the other hand, we show that for some problems, such as computing the parities of rows or columns or deciding if there are two identical rows or columns, quantum computers provide exponential speedup. We demonstrate this by showing equivalence between models that provide matrix-vector products, vector-matrix products, and vector-matrix-vector products, whereas the power of these models can vary significantly for classical computation.

中文翻译：

---

矩阵向量乘积的量子查询复杂度

我们研究使用查询返回输入对向量的作用的查询来学习矩阵属性的量子算法。我们表明，对于各种问题，包括计算矩阵的迹线，行列式或秩或求解其指定的线性系统，量子计算机都不能提供比经典计算更快的渐近速度。另一方面，我们表明，对于某些问题，例如计算行或列的奇偶性或确定是否存在两个相同的行或列，量子计算机提供了指数级加速。我们通过显示提供矩阵向量乘积，向量矩阵乘积和向量矩阵乘积的模型之间的等效性来证明这一点，而对于经典计算，这些模型的功效可能会有很大差异。