Adiabatic Approximation

C. C. Chancey, M. C.M. O’Brien

K. T. Hecht

HuaiXin Cao, Zhihua Guo, ZhengLi Chen +1 more

Juan H. Arredondo

Donny Cheung, Peter Høyer, Nathan Wiebe

The discovery of adiabatic quantum computing has revealed a need for rigorously proven bounds for the error in the adiabatic approximation. We present in this paper, a rigorous and elementary derivation of upper and lower bounds on the error incurred from using the adiabatic approximation for quantum systems that can be used to assess the runtime of adiabatic algorithms. Our bounds are often asymptotically tight in the limit of slow evolution for fixed Hamiltonians, and are used to provide sufficient conditions for the application of the adiabatic approximation. We show that our sufficiency criteria exclude the Marzlin–Sanders counterexample from the class of Hamiltonians that obey the adiabatic approximation. Finally, we demonstrate the existence of classes of Hamiltonians that resemble the Marzlin–Sanders counterexample Hamiltonian, but also obey the adiabatic approximation.

Dafa Li

Abstract The adiabatic theorem was proposed about 90 years ago and has played an important role in quantum physics. The quantitative adiabatic condition constructed from eigenstates and eigenvalues of a Hamiltonian is a traditional tool to estimate adiabaticity and has proven to be the necessary and sufficient condition for adiabaticity. However, recently the condition has become a controversial subject. In this paper, we list some expressions to estimate the validity of the adiabatic approximation. We show that the quantitative adiabatic condition is invalid for the adiabatic approximation via the Euclidean distance between the adiabatic state and the evolution state. Furthermore, we deduce general necessary and sufficient conditions for the validity of the adiabatic approximation by different definitions.