Difference between quantum annealing by imaginary-time and real-time Schrödinger equations of Grover’s search

We confirmed the annealing time required to obtain the desired success probability for quantum annealing by the imaginary-time and real-time Schrödinger equations of Grover’s search, with two kinds of schedulings; one linearly decreases the quantum fluctuation and the other tunes the evolution rate of the Hamiltonian on the basis of the adiabatic condition. With linear scheduling, the required annealing time for quantum annealing by the imaginary-time Schrödinger equation is of order log N, which is very different from O(N) required for that by the real-time Schrödinger equation. With the scheduling based on the adiabatic condition, the required annealing time is of order pN, which is identical to ffiffiffiffi that by the real-time Schrödinger equation. Although the scheduling based on the adiabatic condition is optimal for the quantum annealing by the real-time Schrödinger equation, it is inefficient for that by the imaginary-time Schrödinger equation. This result implies that the optimal schedulings for the quantum annealing by the imaginary-time and real-time Schrödinger equations differ greatly, and the efficient scheduling considered with the quantum Monte Carlo method, which is based on the imaginary-time Schrödinger equation, is not necessarily effective in improving the performance of quantum annealing by the real-time Schrödinger equation. We discuss the efficient scheduling for quantum annealing by the imaginary-time Schrödinger equation with respect to the exponential decay of excited states.