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arXiv · 2508.10361

Saturable Quantum Speed Limits for Imaginary-Time Evolution

Kohei Kobayashi
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Abstract

We derive a Geometric quantum speed limit (QSL) for imaginary-time evolution, where the dynamics is governed by a non-unitary Schr\"{o}dinger equation. By introducing a cost function based on the angular distance between the normalized evolving state and the initial state, we obtain a lower bound on the evolution time expressed as the ratio between this angle and the time-averaged energy dispersion. Our bound is analytical, general, and applicable to arbitrary time-independent Hamiltonians. We analytically evaluate this bound for two physically motivated cases. First, we apply it to a two-level system and derive an expression for the minimal time. Second, we analyze the imaginary-time version of Grover search problem and rigorously reproduce the well-known logarithmic scaling $T=\mathcal{ O}(\log N)$ within our QSL framework.

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Kohei Kobayashi. 2025-08-14. Saturable Quantum Speed Limits for Imaginary-Time Evolution. https://arxiv.org/abs/2508.10361

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