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——例如源自非厄米物理和微分方程的问题——可以通过在量子计算机上处理非正规输入矩阵的特征值来解决。然而，现有的量子奇异值变换（QSVT）框架并不适合此任务，因为特征值和奇异值通常并不相同。我们提出了一个量子特征值变换（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，以及一个相关的量子特征值估计（QEVE）算法，用于处理具有实数谱的算子。QEVT对块编码的查询复杂度几乎恢复了QSVT在厄米输入情况下的复杂度，而QEVE对于可对角化输入矩阵实现了海森堡极限标度。作为应用，我们开发了一个线性微分方程求解器，对于平均情况下的可对角化算子具有严格的线性时间查询复杂度，以及一个基态制备算法，该算法将先前针对厄米哈密顿量的近乎最优结果提升至具有实数谱的可对角化矩阵。我们算法的基础是一种制备法伯多项式量子叠加态的有效方法，该方法将切比雪夫多项式在复平面上的近乎最佳一致逼近性质进行了推广。作为独立贡献，我们还开发了以 $\mathbf{O}(\mathrm{polylog}(n))$ 门生成 $n$ 个傅里叶系数的技术，而先前方法的成本是线性的。