Rapid initial state preparation for the quantum simulation of strongly correlated molecules and materials

Studies on quantum algorithms for ground state energy estimation often assume perfect ground state preparation; however, in reality the initial state will have imperfect overlap with the true ground state. Here we address that problem in two ways: by faster preparation of matrix product state (MPS) approximations, and more efficient filtering of the prepared state to find the ground state energy. We show how to achieve unitary synthesis with a Toffoli complexity about $7 \times$ lower than that in prior work, and use that to derive a more efficient MPS preparation method. For filtering we present two different approaches: sampling and binary search. For both we use the theory of window functions to avoid large phase errors and minimise the complexity. We find that the binary search approach provides better scaling with the overlap at the cost of a larger constant factor, such that it will be preferred for overlaps less than about 0.003. Finally, we estimate the total resources to perform ground state energy estimation of FeMoco and Iron cluster systems by estimating ground state overlap on an MPS initial state through extrapolation. With a modest bond dimension of 4000 we estimate a 0.96 overlap squared value producing total resources of $7.5 \times 10^{10}$ Toffoli gates; validating naive estimates where we assume perfect ground state overlap. These extrapolations allay practical concerns of exponential overlap decay in challenging-to-compute chemical systems.