Error Estimates For The Quantum Adiabatic Approximation - Nathan Wiebe

The adiabatic approximation is frequently used in quantum sciences to approximate the time-evolution of quantum systems that are governed by a slowly varying time-dependent Hamiltonian. Despite the ubiquity of the approximation, it was shown in 2004 by Peter Marzlin and Barry Sanders that the criterion that is most commonly used for determining when the approximation is valid can fail for certain Hamiltonians. We address these issues by presenting an intuitive and elementary proof of the adiabatic approximation that is based on path integrals. Furthermore, we provide the tightest upper and lower bounds on the error in the adiabatic approximation that are currently known, and show that the bounds are asymptotically tight for many Hamiltonians. Finally, we use these bounds to show that the Marzlin-Sanders counterexample Hamiltonian is marginally pathological in that if its second time-derivative were slightly diminished then the Hamiltonian would no longer be a counterexample. This work is collaborative work done with Donny Cheung and Peter Hoyer.