Proof of the Adiabatic Theorem

Andris Ambainis, Oded Regev

We provide an elementary proof of the quantum adiabatic theorem.

Rachel Berkowitz

M. S. Sarandy, Lian-Ao Wu, Daniel A. Lidar

Yan Zhao

The quantum adiabatic theorem is a widely applied result in quantum mechanics. However, recently, counterexamples were produced, indicating that the transition probabilities in some systems do not converge as predicted by the adiabatic theorem [K. P. Marzlin and B. C. Sanders, Phys. Rev. Lett. 93, 160408 (2004)]. In this paper, we examine a proof of the adiabatic theorem, paying specific attention to the validity of the assumptions made in the proof. We show that the convergence of the transition probabilities is path dependent when the change in the Hamiltonian is large. Finally, we consider the application of this reformulated form of the adiabatic theorem to the adiabatic evolution paradigm in quantum computation.

Mirjana Božić, R.J. Lombard, Z. Marić

Andrew Das Arulsamy

An explicit proof is developed to reinforce the accuracy of the quantum adiabatic theorem in its original form without any inconsistency and/or violation. Based on this proof, we discuss physical implications that give rise to the violation of the quantum adiabatic approximation. We show that such a violation can be obtained if and only if one violates the adiabatic criterion itself or due to the existence of degeneracy at a later time. Subsequently, comparison of our proof with respect to other recently developed proofs and counter-examples are analyzed and discussed.