Resolvent-based quantum phase estimation: Towards estimation of parametrized eigenvalues

Quantum algorithms for estimating the eigenvalues of matrices, including the phase estimation algorithm, serve as core subroutines in a wide range of quantum algorithms, including those in quantum chemistry and quantum machine learning. In standard quantum eigenvalue (phase) estimation, a Hermitian (unitary) matrix and a state in an unknown superposition of its eigenstates are provided, with the objective of estimating and coherently recording the corresponding real eigenvalues (eigenphases) in an ancillary register. Estimating eigenvalues of non-normal matrices presents unique challenges, as the eigenvalues may lie anywhere on the complex plane. Furthermore, the non-orthogonality of eigenvectors and the existence of generalized eigenvectors complicate the implementation of matrix functions. In this work, we propose a novel approach for estimating the eigenvalues of non-normal matrices based on the matrix resolvent formalism. We construct the first efficient algorithm for estimating the phases of the unit-norm eigenvalues of a given non-unitary matrix. We then construct an efficient algorithm for estimating the real eigenvalues of a given non-Hermitian matrix, achieving complexities that match the best known results while operating under significantly relaxed assumptions on the non-real part of the spectrum. The resolvent-based approach that we introduce also extends to estimating eigenvalues that lie on a parametrized complex curve, subject to explicitly stated conditions, thereby paving the way for a new paradigm of parametric eigenvalue estimation.