  The goal of digital quantum simulation is to approximate the dynamics of a
given target Hamiltonian via a sequence of quantum gates, a procedure known as
Trotterization. The quality of this approximation can be controlled by the so
called Trotter step, that governs the number of required quantum gates per unit
simulation time. The stroboscopic dynamics generated by Trotterization is
effectively described by a time-independent Hamiltonian, referred to as the
Floquet Hamiltonian. In this work, we propose Floquet Hamiltonian learning to
reconstruct the experimentally realized Floquet Hamiltonian order-by-order in
the Trotter step. This procedure is efficient, i.e., it requires a number of
measurements that scales polynomially in the system size, and can be readily
implemented in state-of-the-art experiments. With numerical examples, we
propose several applications of our method in the context of verification of
quantum devices: from the characterization of the distinct sources of errors in
digital quantum simulators to determining the optimal operating regime of the
device. We show that our protocol provides the basis for feedback-loop design
and calibration of new types of quantum gates. Furthermore it can be extended
to the case of non-unitary dynamics and used to learn Floquet Liouvillians,
thereby offering a way of characterizing the dissipative processes present in
NISQ quantum devices.
