A Geometric Operator Quantum Speed Limit

By formulating operator flows in the framework of quantum geometry, we derive a MT-type OQSL [1] that constrains the operator growth in isolated quantum systems undergoing arbitrarily driven dynamics. Differently from the OQSLs introduced in Chap.  5 , this result applies to arbitrary time-dependent Hamiltonians. The geodesic argument underlying the derivation of the OQSL allows us to precisely determine the conditions for the saturation of the bound. We further apply the geometric OQSL to Hamiltonian flows and to the evolution of Krylov complexity, establishing a direct comparison between the formulation of operator growth in Krylov space and the framework of OQSLs. Remarkably, the two approaches identify the same notion of tightness and set a unique timescale on the dynamics of operator growth.