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We iteratively apply a recently formulated adiabatic theorem for the strong-coupling limit in finite-dimensional quantum systems. This allows us to improve approximations to a perturbed dynamics, beyond the standard approximation based on quantum Zeno dynamics and adiabatic elimination. The effective generators describing the approximate evolutions are endowed with the same block structure as the unperturbed part of the generator, and exhibit adiabatic evolutions. This iterative adiabatic theorem reveals that adiabaticity holds eternally, that is, the system evolves within each eigenspace of the unperturbed part of the generator, with an error bounded by $O(1/gamma)$ uniformly in time, where $gamma$ characterizes the strength of the unperturbed part of the generator. We prove that the iterative adiabatic theorem reproduces Blochs perturbation theory in the unitary case, and is therefore a full generalization to open systems. We furthermore prove the equivalence of the Schrieffer-Wolff and des Cloiseaux approaches in the unitary case and generalize both to arbitrary open systems, showing that they share the eternal adiabaticity, and providing explicit error bounds. Finally we discuss the physical structure of the effective adiabatic generators and show that ideal effective generators for open systems do not exist in general.
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A combination of the digitized shortcut-to-adiabaticity (STA) and the sequential digitized adiabaticity is implemented in a superconducting quantum device to determine electronic states in two example systems, the H2 molecule and the topological Bernevig-Hughes-Zhang (BHZ) model. For H2, a short internuclear distance is chosen as a starting point, at which the ground and excited states are obtained via the digitized STA. From this starting point, a sequence of internuclear distances is built. The eigenstates at each distance are sequentially determined from those at the previous distance via the digitized adiabaticity, leading to the potential energy landscapes of H2. The same approach is applied to the BHZ model, and the valence and conduction bands are excellently obtained along the X-{Gamma}-X linecut of the first Brillouin zone. Furthermore, a numerical simulation of this method is performed to successfully extract the ground states of hydrogen chains with the lengths of 3 to 6 atoms.
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