A useful fundamental speed limit for the imaginary-time Schrodinger equation

The quantum speed limit (QSL), or the energy-time uncertainty relation, gives a fundamental speed limit for quantum dynamics. Recently, Kieu [arXiv:1702.00603] derived a new class of QSL which is not only formal but also suitable for actually evaluating the speed limit. Inspired by his work, we obtain a similar speed limit for the imaginary-time Schr\"odinger equation. Using this new bound, we show that the optimal computational time of the Grover problem in imaginary-time quantum annealing is bounded from below by $\log N$, which is consistent with a result of previous study.