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Choosing an optimal time step $\delta t$ is crucial for an efficientHamiltonian simulation based on Trotterization but difficult due to the complexstructure of the Trotter error. Here we develop a method measuring the Trottererror by combining the second- and fourth-order Trotterizations rather thanconsulting with mathematical error bounds. Implementing this method, weconstruct an algorithm, which we name Trotter24, for adaptively using almostthe largest stepsize $\delta t$, which keeps quantum circuits shallowest,within an error tolerance $\epsilon$ preset for our purpose. Trotter24 appliesto generic Hamiltonians, including time-dependent ones, and can be generalizedto any orders of Trotterization. Benchmarking it in a quantum spin chain, wefind the adaptively chosen $\delta t$ to be about ten times larger than thatinferred from known upper bounds of Trotter errors. Trotter24 allows us to keepthe quantum circuit thus shallower within the error tolerance in exchange forpaying the cost of measurements.

Trotter24: A precision-guaranteed adaptive stepsize Trotterization for Hamiltonian simulations