Optimal Quantum Eigenvalue Transformation via Linear Combinations of Hermitian Matrices

How can we efficiently implement general functions of non-normal matrices on a quantum computer? While Quantum Singular Value Transformation (QSVT) is highly effective for Hermitian matrices, applying it to non-normal matrices is significantly more complex due to the potential for leakage out of the success subspace and the dependence on the matrix's Jordan structure.

The authors develop two complementary LCHM formulations:

1. Vanilla LCHM: A generalization of existing Hamiltonian simulation techniques that represents matrix functions as kernel integrals of Hermitian pencils, particularly useful for entire or half-plane analytic functions.
2. Weyl LCHM: An exact angular projection method that expresses matrix powers and other functions as an integral of Chebyshev polynomials of angular Hermitian matrices. This approach allows for an exact discrete implementation using root-of-unity sampling, eliminating truncation and quadrature errors.

These formulations are integrated into a coherent Quantum Eigenvalue Transformation (QET) algorithm. By using a lifted scalar function approach, the authors show that uniform scalar approximation error translates directly to the operator norm with constant-one scaling.

The study demonstrates that Weyl LCHM provides a degree-optimal $\Theta(d)$ circuit depth for a degree-$d$ polynomial transformation. It also achieves optimal post-selection repetition counts, scaling as $\mathcal{O}(\|p_d\|_{\infty}/\|p_d(A)|\psi\rangle\|)$. The method is highly versatile, unifying various quantum linear algebraic tasks including matrix exponentials, resolvents, logarithms, and fractional powers. Furthermore, the authors introduce a Faber-Weyl variant that adapts to non-circular numerical ranges, maintaining optimal complexity while providing a robust framework for general convex domains.