《工程》是中国工程院(CAE)于2015年推出的国际开放存取期刊。其目的是提供一个高水平的平台,传播和分享工程研发的前沿进展、当前主要研究成果和关键成果;报告工程科学的进展,讨论工程发展的热点、兴趣领域、挑战和前景,在工程中考虑人与环境的福祉和伦理道德,鼓励具有深远经济和社会意义的工程突破和创新,使之达到国际先进水平,成为新的生产力,从而改变世界,造福人类,创造新的未来。 期刊链接:https://www.sciencedirect.com/journal/engineering

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Kernel matrices, which arise from discretizing a kernel function $k(x,x')$, have a variety of applications in mathematics and engineering. Classically, the celebrated fast multipole method was designed to perform matrix multiplication on kernel matrices of dimension $N$ in time almost linear in $N$ by using techniques later generalized into the linear algebraic framework of hierarchical matrices. In light of this success, we propose a quantum algorithm for efficiently performing matrix operations on hierarchical matrices by implementing a quantum block-encoding of the hierarchical matrix structure. When applied to many kernel matrices, our quantum algorithm can solve quantum linear systems of dimension $N$ in time $O(\kappa \operatorname{polylog}(\frac{N}{\varepsilon}))$, where $\kappa$ and $\varepsilon$ are the condition number and error bound of the matrix operation. This runtime is exponentially faster than any existing quantum algorithms for implementing dense kernel matrices. Finally, we discuss possible applications of our methodology in solving integral equations or accelerating computations in N-body problems.

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