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The state $|\psi(t)\rangle$ of a closed quantum system evolves under theSchr\"{o}dinger equation, where the reversible evolution of the state isdescribed 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 innon-reversible evolutions, described by Lindblad equations. The solution ofthese equations give rise to quantum channels $\Phi_t$ that describe theevolution of density matrices according to $\rho(t)=\Phi_t(\rho_0)$, whichoften results in decoherence and dephasing of the state. For many quantumexperiments, the time until which measurements can be done might be limited,e.g. by experimental instability or technological constraints. However, furtherevolution of the state may be of interest. For instance, to determine thesource of the decoherence and dephasing, or to identify the steady state of theevolution. In this work, we introduce a method to approximate a given targetquantum channel by means of variationally approximating equivalent unitaries onan extended system, invoking the Stinespring dilation theorem. We report on anexperimentally feasible method to extrapolate the quantum channel on discretetime steps using only data on the first time steps. Our approach heavily relieson the ability to spatially transport entangled qubits, which is unique to theneutral atom quantum computing architecture. Furthermore, the method showspromising predictive power for various non-trivial quantum channels. Lastly, aquantitative analysis is performed between gate-based and pulse-basedvariational quantum algorithms.

Variational method for learning Quantum Channels via Stinespring Dilation on neutral atom systems