Bounds on a Wavefunction Overlap with Hamiltonian Eigen-states: Performance Guarantees for the Quantum Phase Estimation Algorithm

Estimating the overlap between an approximate wavefunction and a target eigenstate of the system Hamiltonian is essential for the efficiency of quantum phase estimation. In this work, we derive upper and lower bounds on this overlap using expectation values of Hamiltonian powers and bounds on target eigenenergies. The accuracy of these bounds can be systematically improved by computing higher-order Hamiltonian moments and refining eigenenergy estimates. Our method offers a practical approach to assessing initial state quality and can be implemented on both classical and quantum computers.