A note on the switching adiabatic theorem (Q2872285)

The paper ``A note on the switching adiabatic theorem'' by A. Elgart and G. A. Hagedorn bounds the running time for the adiabatic theorem with smoothness conditions on the time-dependent Hamiltonian and the minimum energy gap \(g\) between the eigenvalue of interest and the rest of the spectrum. Formally, when the time-dependent Hamiltonian belongs to the Gevrey class \(G^\alpha\), the time-dependent error in the adiabatic approximation remains small for total runtimes scaling as \(g^{-2} |\ln g|^{6\alpha}\). Here, the smoothness conditions on the Hamiltonian add logarithmic corrections on the well-known Landau-Zener result. NEWLINENEWLINENEWLINEIn their introduction, the authors briefly review the Schrödinger equation and practical applications such as adiabatic quantum computation. Then, they state their main result, Theorem 1.2. Furthermore, they introduce Gevrey classes and provide a corresponding example, relate with previous work and outline the proof strategy for their main result. In Sec. II, Nenciu's expansion from Ref. 24 is presented together with a formal solution. In Sec. III, the authors bound the terms in the Nenciu expansion with the minimum energy gap \(g\). Finally, in Sec. IV, these bounds are used to prove their main result. The paper contains an appendix containing technical results. NEWLINENEWLINENEWLINECompared to the direct solution of the Schödinger equation e.g. with time-dependent energy eigenstates, Nenciu's expansion is a very elegant tool, which might also have the potential to be generalized to the evolution of open systems. The field of adiabatic computation will probably not immediately profit from the main result, as for adiabatic algorithms the known \(g^{-2}\)-runtime scaling is the main bottleneck. The paper is densely written and contains all technical details, although in parts it might have profited from some more application examples. Therefore, it will be very useful for specialists interested in rigorous bounds on the adiabatic runtime.