Bounding Eigenstate Overlap from Hamiltonian Moments: Success Probability Guarantees for Quantum Phase Estimation

Estimating the overlap between a prepared state and a target eigenstate is crucial for the efficiency of quantum phase estimation (QPE), since QPE succeeds with probability equal to this overlap. We present a systematically improvable method to compute certified upper and lower bounds on such overlaps using a finite set of Hamiltonian moments. Our approach constructs optimal polynomial upper/lower bounds on an energy-window indicator and evaluates them through linear and semidefinite programs, yielding the tightest bounds consistent with the available moment and spectral-interval information. We demonstrate the method on strongly correlated molecular Hamiltonians and study the impact of approximate moments obtained from tensor-network contractions. The resulting bounds provide a practical pre-QPE screening tool for selecting initial states and can be implemented with either classical moment computation or quantum expectation estimation.