  We confirmed the annealing time of Grover's search which is required to
obtain desired success probability for quantum annealing by the imaginary-time
and the real-time Schr\"{o}dinger equation with two kinds of schedulings; one
linearly decreases the quantum fluctuation and the other tunes the evolution
rate of the Hamiltonian based on the adiabatic condition. With linear
scheduling, the required annealing time for quantum annealing by the
imaginary-time Schr\"{o}dinger equation is of order $\log N$, which is very
different from $O(N)$ required for the quantum annealing by the real-time
Schr\"{o}dinger equation. With the scheduling based on the adiabatic condition,
the required annealing time is of order $\sqrt{N}$, which is identical to the
annealing time for quantum annealing by the real-time Schr\"{o}dinger equation.
Although the scheduling based on the adiabatic condition is optimal for the
quantum annealing by the real-time Schr\"{o}dinger equation, it is inefficient
for the quantum annealing by the imaginary-time Schr\"{o}dinger equation. This
result implies that the optimal scheduling for the quantum annealing by the
imaginary-time and the real-time Schr\"{o}dinger equation is very different,
and the efficient scheduling considered with the quantum Monte Carlo methods,
which is based on imaginary-time Schr\"{o}dinger equation, is not necessarily
effective to improve the performance of quantum annealing by the real-time
Schr\"{o}dinger equation. We discuss the efficient scheduling for quantum
annealing by the imaginary-time Schr\"{o}dinger equation with respect to the
exponential decay of excited states.
