Minimum Fourth-Order Trotterization Formula for a Time-Dependent Hamiltonian

When a time propagator $e^{\delta t A}$ for duration $\delta t$ 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 $A(t)$. Its difficulty is that the propagator becomes a time-ordered exponential $\mathcal{T}\exp(\int_{\mu-\delta t/2}^{\mu+\delta t/2}A(s)ds)$ for more than the second-order formula. 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 obtain a fourth-order Trotterization formula, whose error is $O(\delta t^5)$. The formula consists of seven exponentials of $X$ and $Y$, and we prove that there is no fourth-order Trotterization formula with fewer than seven exponentials. Its error consists of the contribution $\Gamma_5$ known for the time-independent formula plus a new contribution $\Upsilon_5$ which is intrinsic to the time dependence of $A(t)$. Finally, we numerically demonstrate that for the Hamiltonian tested our formula has errors as small as the time-dependent fourth-order Suzuki formula involving eleven exponentials.