Eigenvalue transformations, which include solving time-dependent differential equations as a special case, have a wide range of applications in scientific and engineering computation. While quantum algorithms for singular value transformations are well studied, eigenvalue transformations are distinct, especially for non-normal matrices. We propose an efficient quantum algorithm for performing a class of eigenvalue transformations that can be expressed as a certain type of matrix Laplace transformation. This allows us to significantly extend the recently developed linear combination of Hamiltonian simulation (LCHS) method [An, Liu, Lin, Phys. Rev. Lett. 131, 150603, 2023; An, Childs, Lin, arXiv:2312.03916] to represent a wider class of eigenvalue transformations, such as powers of the matrix inverse, $A^{-k}$, and the exponential of the matrix inverse, $e^{-A^{-1}}$. The latter can be interpreted as the solution of a mass-matrix differential equation of the form $A u'(t)=-u(t)$. We demonstrate that our eigenvalue transformation approach can solve this problem without explicitly inverting $A$, reducing the computational complexity.

翻译：特征值变换（包括求解时间相关微分方程作为特例）在科学与工程计算中具有广泛应用。尽管奇异值变换的量子算法已得到深入研究，但特征值变换（特别是针对非正规矩阵的情形）具有本质区别。本文提出一种高效的量子算法，用于执行可表示为特定类型矩阵拉普拉斯变换的特征值变换。这使我们能够显著扩展近期发展的哈密顿模拟线性组合（LCHS）方法[An, Liu, Lin, Phys. Rev. Lett. 131, 150603, 2023; An, Childs, Lin, arXiv:2312.03916]，以表示更广泛类别的特征值变换，例如矩阵逆的幂次 $A^{-k}$ 和矩阵逆的指数函数 $e^{-A^{-1}}$。后者可解释为形如 $A u'(t)=-u(t)$ 的质量矩阵微分方程的解。我们证明所提出的特征值变换方法可在无需显式求逆 $A$ 的情况下求解该问题，从而降低计算复杂度。