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We confirmed the annealing time of Grover's search which is required toobtain desired success probability for quantum annealing by the imaginary-timeand the real-time Schr\"{o}dinger equation with two kinds of schedulings; onelinearly decreases the quantum fluctuation and the other tunes the evolutionrate of the Hamiltonian based on the adiabatic condition. With linearscheduling, the required annealing time for quantum annealing by theimaginary-time Schr\"{o}dinger equation is of order $\log N$, which is verydifferent from $O(N)$ required for the quantum annealing by the real-timeSchr\"{o}dinger equation. With the scheduling based on the adiabatic condition,the required annealing time is of order $\sqrt{N}$, which is identical to theannealing time for quantum annealing by the real-time Schr\"{o}dinger equation.Although the scheduling based on the adiabatic condition is optimal for thequantum annealing by the real-time Schr\"{o}dinger equation, it is inefficientfor the quantum annealing by the imaginary-time Schr\"{o}dinger equation. Thisresult implies that the optimal scheduling for the quantum annealing by theimaginary-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 necessarilyeffective to improve the performance of quantum annealing by the real-timeSchr\"{o}dinger equation. We discuss the efficient scheduling for quantumannealing by the imaginary-time Schr\"{o}dinger equation with respect to theexponential decay of excited states.

Difference between quantum annealing by imaginary-time and real-time Schr\"{o}dinger equation of Grover's search