Minimum Trotterization Formulas for a Time-Dependent Hamiltonian

When a time propagator e^{δt A} for duration δt consists of two noncommuting parts A=X+Y, Trotterization approximately decomposes the propagator into a product of exponentials of X and Y. Various Trotterization formulas have been utilized in quantum and classical computers, but much less is known for the Trotterization with the time-dependent generator 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 formulas with minimum possible exponentials. In particular, we obtain fourth-order and sixth-order Trotterization formulas involving seven and fifteen exponentials, respectively, 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 numerically benchmark the fourth-order formulas 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.