Saturable Quantum Speed Limits for Imaginary-Time Evolution

TL;DR

This paper establishes a geometric quantum speed limit for imaginary-time evolution, providing a universal lower bound on evolution time applicable to various Hamiltonians, with analytical results for specific quantum systems.

Contribution

It introduces a new geometric bound for imaginary-time evolution, extending quantum speed limits to non-unitary dynamics and analytically evaluating it for key quantum models.

Findings

Derived a universal lower bound on imaginary-time evolution time.

Applied the bound to two-level systems to find minimal evolution times.

Reproduced the logarithmic scaling in Grover search within the new QSL framework.

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.