Optimal Quantum Eigenvalue Transformation via Linear Combinations of Hermitian Matrices

We discover two complementary linear-combination-of-Hermitian-matrices (LCHM) formulations to achieve a general non-normal matrix eigenvalue transformation $g(A)$. Firstly, for $A=L+\mathrm{i} H$ with Hermitian $L$ and $H$, the vanilla LCHM formula represents $g(A)$ as a kernel integral of $g(\mathrm{i}(H+kL))$, and it contains linear-combination-of-Hamiltonian-simulation (LCHS) [An, Liu, Lin, Phys. Rev. Lett. 2023] as the special case for matrix exponentials. Secondly, for the angular Hermitian $X_\theta = \cos\theta L+\sin\theta H$, the Weyl LCHM formula expresses $g(A)$ via integrating $g(\mathrm{e}^{\mathrm{i}\theta} (X_\theta\pm\mathrm{i}(I-X_\theta^2)^{1/2}))$. For the matrix power $g(A)=A^m$, the Fourier projection of Weyl LCHM gives $$A^m=\frac{2}{\pi}\int_0^\pi \text{e}^{\text{i} m\theta}T_m(X_\theta) \text{d}\theta = \frac{2}{N}\sum_{j=0}^{N-1} \text{e}^{\text{i} m\theta_j}T_m(X_{\theta_j}),\quad\theta_j=\frac{\pi j}{N},\quad \text{for every } N>m$$ with Chebyshev polynomial of Hermitian $T_m(X_\theta)$ and $N$ samples. The discrete formula is exact, introduces no truncation and angular quadrature error, and offers $\mathcal{O}(1)$ post-selection weights. LCHM formulas lead to new quantum eigenvalue transformation (QET) algorithms. For a degree-$d$ polynomial $p_d(A)$ on $|\psi\rangle$, our QET algorithm can achieve optimal $\Theta(d)$ circuit depth and optimal $\mathcal{O}(||p_d||_{\infty}/||p_d(A)|\psi\rangle||)$ post-selection repetitions. LCHM-based QETs unify various quantum linear algebraic problems with near-optimal $\mathcal{\widetilde O}(d\log(d/\epsilon))$ Clifford$+T$ gates, including driven ODEs (reduced to standard LCHS), iterative methods, resolvents, $\log(I+A)$, $(\lambda I+A)^\nu$, Sign and ReLU transforms, and Faber approximation on noncircular domains.