Difference between Quantum Annealing by Imaginary-Time and Real-Time Schrödinger Equations of Grover’s Search

Abstract

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 Schrödinger 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 Schrödinger equation is of order $\log N$, which is very different from $O(N)$ required for the quantum annealing by the real-time Schrödinger 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 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 the quantum annealing by the imaginary-time Schrödinger equation. This result implies that the optimal scheduling for the quantum annealing by the imaginary-time and the real-time Schrödinger equation is very different, and the efficient scheduling considered with the quantum Monte Carlo methods, which is based on imaginary-time Schrödinger equation, is not necessarily effective to improve 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.