On the time-independent Hamiltonian in real-time and imaginary-time quantum annealing**Project supported by the China Postdoctoral Science Foundation (Grant No. 2017M620322), the Priority Fund for the Postdoctoral Scientific and Technological Program of Hubei Province in 2017, the Seed Foundation of Huazhong University of Science and Technology (Grant No. 2017KFYXJJ070), and the Science and Technology Program of Shenzhen of China (Grant No.

Abstract

We present the analog analogue of Grover’s problem as an example of the time-independent Hamiltonian for applying the speed limit of the imaginary-time Schrödinger equation derived by Okuyama and Ohzeki and the new class of energy-time uncertainty relation proposed by Kieu. It is found that the computational time of the imaginary-time quantum annealing of this Grover search can be exponentially small, while the counterpart of the quantum evolution driven by the real-time Schrödinger equation could only provide square root speedup, compared with classic search. The present results are consistent with the cases of the time-dependent quantum evolution of the natural Grover problem in previous works. We once again emphasize that the logarithm and square root algorithmic performances are generic in imaginary-time quantum annealing and quantum evolution driven by real-time Schrödinger equation, respectively. Also, we provide evidences to search deep reasons why the imaginary-time quantum annealing can lead to exponential speedup and the real-time quantum annealing can make square root speedup.