Many standard linear algebra problems can be solved on a quantum computer by using recently developed quantum linear algebra algorithms that make use of block encodings and quantum eigenvalue/singular value transformations. A block encoding embeds a properly scaled matrix of interest A in a larger unitary transformation U that can be decomposed into a product of simpler unitaries and implemented efficiently on a quantum computer. Although quantum algorithms can potentially achieve exponential speedup in solving linear algebra problems compared to the best classical algorithm, such gain in efficiency ultimately hinges on our ability to construct an efficient quantum circuit for the block encoding of A, which is difficult in general, and not trivial even for well-structured sparse matrices. In this paper, we give a few examples on how efficient quantum circuits can be explicitly constructed for some well-structured sparse matrices, and discuss a few strategies used in these constructions. We also provide implementations of these quantum circuits in MATLAB.

翻译：许多标准线性代数问题可通过近期发展的量子线性代数算法在量子计算机上求解，这些算法利用了块编码和量子特征值/奇异值变换技术。块编码将适当缩放的目标矩阵A嵌入到更大的酉变换U中，该酉变换可分解为更简单酉算子的乘积，并能在量子计算机上高效实现。尽管量子算法在解决线性代数问题时理论上可比最佳经典算法实现指数级加速，但这种效率提升最终取决于我们能否为A构造高效的块编码量子电路——这一任务在一般情况下十分困难，即便对于结构良好的稀疏矩阵也非易事。本文给出若干示例，展示如何为特定结构化稀疏矩阵显式构造高效量子电路，并讨论这些构造中使用的几种策略。此外，我们还在MATLAB中实现了这些量子电路。