Classical Bounds for Quantum Overlaps: Estimating Wavefunction Overlap for Efficient Quantum Simulation

Estimating the overlap between a target wavefunction and unknown Hamiltonian eigenstates is crucial for both theoretical research and practical applications in quantum computing. Specifically, preparing initial states with high overlap with the target Hamiltonian eigenstates is essential for the efficiency of quantum simulation algorithms, such as those based on quantum phase estimation. Being able to estimate lower or upper bounds on this overlap before executing the algorithm on quantum hardware can enable pre-screening of initial states, reducing the required quantum resources. In this talk, I will present a classical algorithm designed to obtain increasingly accurate lower and upper bounds on the exact overlap between an initial state and arbitrary Hamiltonian eigenstates, given moments of the Hamiltonian with respect to the initial state. Our method generalizes Eckart's lower bound theory and leverages linear programming techniques, making it computationally efficient for classical computers. I will also show numerical results on molecular Hamiltonians, demonstrating the bounds our method provides for overlaps with specific eigenstates, and discuss its scalability for larger systems.