Hamiltonian dynamics simulation using linear combination of unitaries on an ion trap quantum computer

The 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-intensive to implement on the current quantum computers. In this work, we demonstrate LCU implementations on an ion trap quantum computer for calculating squared overlaps $\mathrm{\mid }⟨\psi (t=0)\mathrm{\mid }\psi (t>0)⟩{\mathrm{\mid }}^2$ of time-evolved states. This is achieved by an optimized LCU method, based on pre-selecting relevant unitaries, coupled with a compilation strategy which makes use of quantum multiplexor gates, leading to a significant reduction in the depth and number of two-qubit gates in circuits. For $L$ Pauli strings in a Taylor series expanded $n$-qubit-mapped time evolution operator, we find a two-qubit gate count of $2^{\lceil lo{g}_2(L)\rceil }(2n+1)-n-2$. We test this approach by simulating a Rabi-Hubbard Hamiltonian.