Success Rate Versus Finite Run Time in Local Adiabatic Quantum Search Algorithm

In adiabatic quantum algorithm, the success rate is approximately equal to 1 while the run time satisfies the adiabatic condition. But the relation between the short run time and success rate for adiabatic quantum algorithm is poorly understood. In this paper, we study the success rate of local adiabatic quantum search algorithm with an arbitrary finite run time (non-adiabatic evolution). By solving the time-independent Schrödinger equation, we obtain differential equations to calculate the success rate. The differential equations show that the success rate is closely related to the adiabatic parameter $s\left(t\right)$. Utilize the differential equations, we give the function of success rate versus run time in local adiabatic search numerically. The result indirectly verifies that $T\sim O\left(\sqrt{N}\right)$ is optimal in local adiabatic search. (paper)