Minimum Trotterization Formulas for a Time-Dependent Hamiltonian

When a time propagator e delta tA for dura-tion delta t consists of two noncommuting parts A = X + Y, Trotterization approximately decomposes the propagator into a prod-uct of exponentials of X and Y. Vari-ous Trotterization formulas have been uti-lized in quantum and classical comput-ers, but much less is known for the Trot-terization with the time-dependent gen-erator A(t). Here, for A(t) given by the sum of two operators X and Y with time-dependent coefficients A(t) = x(t)X + y(t)Y, we develop a systematic approach to derive high-order Trotterization for-mulas with minimum possible exp onen-tials. In particular, we obtain fourth-order and sixth-order Trotterization formulas in-volving seven and fifteen exponentials, re-spectively, which are no more than those for time-independent generators. We also construct another fourth-order formula consisting of nine exponentials having a smaller error coefficient. Finally, we nu-merically benchmark the fourth-order for-mulas in a Hamiltonian simulation for a quantum Ising chain, showing that the 9 -exponential formula accompanies smaller errors per local quantum gate than the well-known Suzuki formula.