On the time-independent Hamiltonian in real-time and imaginary-time quantum annealing

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.