Many problems in linear algebra -- such as those arising from non-Hermitian physics and differential equations -- can be solved on a quantum computer by processing eigenvalues of the non-normal input matrices. However, the existing Quantum Singular Value Transformation (QSVT) framework is ill-suited to this task, as eigenvalues and singular values are different in general. We present a Quantum EigenValue Transformation (QEVT) framework for applying arbitrary polynomial transformations on eigenvalues of block-encoded non-normal operators, and a related Quantum EigenValue Estimation (QEVE) algorithm for operators with real spectra. QEVT has query complexity to the block encoding nearly recovering that of the QSVT for a Hermitian input, and QEVE achieves the Heisenberg-limited scaling for diagonalizable input matrices. As applications, we develop a linear differential equation solver with strictly linear time query complexity for average-case diagonalizable operators, as well as a ground state preparation algorithm that upgrades previous nearly optimal results for Hermitian Hamiltonians to diagonalizable matrices with real spectra. Underpinning our algorithms is an efficient method to prepare a quantum superposition of Faber polynomials, which generalize the nearly-best uniform approximation properties of Chebyshev polynomials to the complex plane. Of independent interest, we also develop techniques to generate $n$ Fourier coefficients with $\mathbf{O}(\mathrm{polylog}(n))$ gates compared to prior approaches with linear cost.

翻译：线性代数中的许多问题——例如由非厄米物理和微分方程所产生的问题——可以通过处理非正规输入矩阵的特征值在量子计算机上求解。然而，现有的量子奇异值变换（QSVT）框架并不适用于此任务，因为特征值和奇异值通常不同。我们提出了一种量子特征值变换（QEVT）框架，用于对块编码的非正规算子的特征值应用任意多项式变换，并针对具有实谱的算子设计了一种相关的量子特征值估计算法（QEVE）。QEVT对块编码的查询复杂度几乎恢复了对埃尔米特输入的QSVT复杂度，而QEVE对可对角化的输入矩阵实现了海森堡极限标度。作为应用，我们开发了一种线性微分方程求解器，它对平均情况下的可对角化算子具有严格线性时间的查询复杂度，并提出了一种基态制备算法，将先前对厄米哈密顿量的近乎最优结果推广到具有实谱的可对角化矩阵。支撑我们算法的一种高效方法是制备费伯多项式的量子叠加，该多项式将切比雪夫多项式的近乎最佳一致逼近性质推广到复平面。另外值得关注的是，我们还开发了以 $\mathbf{O}(\mathrm{polylog}(n))$ 个门生成 $n$ 个傅里叶系数的技术，而先前方法则需要线性成本。