Making Trotterization adaptive for NISQ devices and beyond

The digital simulation of quantum many-body dynamics, one of the most promising applications of quantum computers, involves Trotterization as a key element. It is an outstanding challenge to formulate a quantum algorithm allowing for adaptive Trotter time steps. This is particularly relevant for today's noisy intermediate scale quantum devices, where the minimization of the circuit depth is a central optimization task. Here, we introduce an adaptive Trotterization scheme providing a controlled solution of the quantum many-body dynamics of local observables. Our quantum algorithm outperforms conventional fixed-time step Trotterization schemes in a quantum quench and even allows for a controlled asymptotic long-time error, where Trotterized dynamics generically enter challenging regimes. This adaptive method can also be generalized to protect various other kinds of symmetries, which we illustrate by preserving the local Gauss's law in a lattice gauge theory. We discuss the requirements imposed by experimental resources, and point out that our adaptive Trotterization scheme can be of use also in numerical approaches based on Trotterization such as in time-evolving block decimation methods.