  The state $|\psi(t)\rangle$ of a closed quantum system evolves under the
Schr\"{o}dinger equation, where the reversible evolution of the state is
described by the action of a unitary operator $U(t)$ on the initial state
$|\psi_0\rangle$, i.e.\ $|\psi(t)\rangle=U(t)|\psi_0\rangle$. However,
realistic quantum systems interact with their environment, resulting in
non-reversible evolutions, described by Lindblad equations. The solution of
these equations give rise to quantum channels $\Phi_t$ that describe the
evolution of density matrices according to $\rho(t)=\Phi_t(\rho_0)$, which
often results in decoherence and dephasing of the state. For many quantum
experiments, the time until which measurements can be done might be limited,
e.g. by experimental instability or technological constraints. However, further
evolution of the state may be of interest. For instance, to determine the
source of the decoherence and dephasing, or to identify the steady state of the
evolution. In this work, we introduce a method to approximate a given target
quantum channel by means of variationally approximating equivalent unitaries on
an extended system, invoking the Stinespring dilation theorem. We report on an
experimentally feasible method to extrapolate the quantum channel on discrete
time steps using only data on the first time steps. Our approach heavily relies
on the ability to spatially transport entangled qubits, which is unique to the
neutral atom quantum computing architecture. Furthermore, the method shows
promising predictive power for various non-trivial quantum channels. Lastly, a
quantitative analysis is performed between gate-based and pulse-based
variational quantum algorithms.
