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.
翻译:在数学和工程学中,由内核函数 $k(x,x”) 产生的内核矩阵具有多种应用。典型地说,有节制的快速多极法是用来对维度内核矩阵进行矩阵乘法(N$),时间几乎线性地以美元计算。根据这一成功,我们提出了一个量子算法,以便通过对等级矩阵结构进行量子区块编码,高效率地对等级矩阵进行矩阵操作。当应用到许多内核矩阵时,我们的量子算法能够及时(O) (Kapta\ a\ opatorname{poly} (\\krac{Nunvarepsilon}) 美元) 解决维度内内核矩阵的量线性系统乘法($\kappa] 美元(Nfrac{N-varepsilon}) 美元) 。在那里,$\kappappa 和 $\ varepslon 美元是矩阵操作的条件和错误。这个运行时间比任何现有的量子运算法都指数速度快。最后,我们讨论我们的方法在解决N-体问题中解决整体方程式或加速计算问题时可能的应用。