Quantum speed limit as a sensitive probe of Planck-scale effects

Many quantum-gravity scenarios predict a minute modification of the canonical commutator, known as the generalized uncertainty principle (GUP), whose low-energy signatures are, in principle, accessible to state-of-the-art laboratory tests. We compute first-order minimal-length corrections to the quantum speed limit (QSL) for three cases: uniform superpositions in an infinite square well, coherent harmonic-oscillator states, and squeezed-oscillator states. We identify a universal amplification law: for any pure state, the fractional shift of either speed limit scales linearly with $\beta$ and algebraically with the state's effective Hilbert-space size. As the effective Hilbert-space dimension can be exceedingly large, the associated minimal-length signatures are amplified by several orders of magnitude. Using high-precision matter-wave timing data, we set a direct bound on the GUP parameter $\beta$, which quantifies minimal-length quantum-gravity effects. Our analysis indicates that phase-locked, short-time overlap fits on kilogram-scale optical-spring modes can tighten this bound by orders of magnitude. We outline two implementable measurement pipelines -- continuous back-action-evading single-quadrature readout and stroboscopic, phase-locked pulsed tomography -- that exploit this leverage, making QSL-based timing a practical, near-term probe of minimal-length physics on quantum-optical and optomechanical platforms.