TY - JOUR

T1 - Difference between quantum annealing by imaginary-time and real-time Schrödinger equations of Grover’s search

AU - Okada, Shuntaro

AU - Ohzeki, Masayuki

AU - Tanaka, Kazuyuki

N1 - Funding Information:
The authors are deeply grateful to Manaka Okuyama and Jun Takahashi for their constructive comments and discussions, which were essential to our research. One of the authors (M.O.) is grateful for the financial support from JSPS KAKENHI 15H03699 and 16H04382, the JST-START, JST-CREST (No. JPMJCR1402), and the ImPACT program. K.T. was partially supported by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science Grant (No. 18H03303).
Publisher Copyright:
©2019 The Physical Society of Japan

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

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U2 - 10.7566/JPSJ.88.024803

DO - 10.7566/JPSJ.88.024803

M3 - Article

AN - SCOPUS:85060519531

VL - 88

JO - Journal of the Physical Society of Japan

JF - Journal of the Physical Society of Japan

SN - 0031-9015

IS - 2

M1 - 024803

ER -