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以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。块编码将经过适当缩放的感兴趣矩阵 A 嵌入到更大的酉变换 U 中，该酉变换可分解为更简单酉变换的乘积，并能在量子计算机上高效实现。尽管量子算法在求解线性代数问题时相比最佳经典算法可能实现指数级加速，但这种效率提升最终取决于我们能否为矩阵 A 的块编码构建高效的量子电路——这一过程通常极具挑战性，即便是对结构良好的稀疏矩阵也并非易事。本文通过几个示例展示了如何为某些结构良好的稀疏矩阵显式构建高效量子电路，并讨论了这些构建中采用的若干策略。我们还提供了这些量子电路在 MATLAB 中的实现。