Squared overlap calculations with linear combination of unitaries

Linear combination of unitaries (LCU) method has proven to scale better than existing product formulas in simulating long time Hamiltonian dynamics. However, given the number of multi-control gate operations in the standard prepare-select-unprepare architecture of LCU, it is still resource-demanding to implement on the current hardware machines. For a special purpose of calculating squared overlaps of two wave functions, we propose an optimized LCU method, based on pre-selecting relevant unitaries, which leads to a significant reduction in the circuit depths and two-qubit counts. We test this approach in simulating Rabi-Hubbard Hamiltonian.