Improved Trotter formula for time-dependent Hamiltonians

Hamiltonian simulations of time-dependent systems are important for a variety of applications, from chemical reactions to optimization problems. Various simulation algorithms have been proposed for time-independent systems, but far less work has been done on algorithms for time-dependent systems.



What is lacking is an algorithm that incorporates the time variant of the Hamiltonian and a theory to quantify the error of the algorithm. The most common time-dependent simulation algorithm is based on the Lie-Trotter formula. However, it is inefficient because it discretizes the time dependence until each step can be approximated as time-independent. A time-dependent version of the Trotter expansion has also been proposed, but it is not considered an independent algorithm because it requires time-ordered integration even after the expansion. The Magnus and Dyson series expansions have been studied as algorithms that incorporate time variation, but they are inefficient at higher orders because the number of terms required combinatorially increases. Also, there is currently no error formula to compare these algorithms. Although error analysis for Trotter's time step has been done, there is no general formula for the error for each of the terms that make up the Hamiltonian.

In this project, we propose an extended Trotter formula for Hamiltonian simulation in the time-dependent case. This algorithm uses the Taylor expansion of the Hamiltonian so that the time derivative of the Hamiltonian appears in the formula. Then, by constructing a general theory to analyze the Trotter error in the time-dependent case, we show that this algorithm has less gate complexity for the same error than conventional algorithms. Our expansion and error theory provide a construction that allows us to derive the extended Trotter formulas with the smallest error in a given ansatz for arbitrary higher order.