2026/07/28 by Yanqiao Wang, Yixuan Liang, Hongjia Chen +1
#quant-ph
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+i H with Hermitian L and H, the vanilla LCHM formula represents g(A) as a kernel integral of g(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 Xθ= cosθL+sinθH, the Weyl LCHM formula expresses g(A) via integrating g(eiθ (Xθ\pmi(I-Xθ2)1/2)). For the matrix power g(A)=Am, the Fourier projection of Weyl LCHM gives Am=\frac2π∫0πei mθTm(Xθ) dθ= (2)/(N)∑j=0N-1 ei mθjTm(Xθj), θj=(πj)/(N), for every N>m with Chebyshev polynomial of Hermitian Tm(Xθ) and N samples. The discrete formula is exact, introduces no truncation and angular quadrature error, and offers O(1) post-selection weights. LCHM formulas lead to new quantum eigenvalue transformation (QET) algorithms. For a degree-d polynomial pd(A) on |ψ⟩, our QET algorithm can achieve optimal Θ(d) circuit depth and optimal O(||pd||∞/||pd(A)|ψ⟩||) post-selection repetitions. LCHM-based QETs unify various quantum linear algebraic problems with near-optimal \widetilde O(dlog(d/ε)) Clifford+T gates, including driven ODEs (reduced to standard LCHS), iterative methods, resolvents, log(I+A), (λI+A)ν, Sign and ReLU transforms, and Faber approximation on noncircular domains.