Quantum speed limits to operator growth

The time evolution of isolated quantum systems can be assessed by formulating constraints on their unitary dynamics. In this sense, quantum speed limits (QSLs) identify fundamental timescales of physical processes by providing lower bounds on the time of evolution between distinguishable quantum states. In this thesis, we introduce universal speed limits to the growth of quantum complexity and to the fundamental speed of a physical process. Motivated by the observation that the dynamics of many-body systems can be better unraveled in the Heisenberg picture, we specifically address the unitary evolution of quantum observables, that is, the so-called operator growth. In the Krylov space representation, an initial operator becomes increasingly complex with the passing of time, a feature that can be quantified by the Krylov complexity. By means of a generalized uncertainty relation involving the generator of the dynamics, we formulate a universal speed limit to the growth of the Krylov complexity that we refer to as the dispersion bound \cite{Hornedal22}. We show the conditions for this bound to be saturated and illustrate its validity in paradigmatic models of quantum chaos. Remarkably, we find that the maximal speed limit can be reached even by a qubit, that is, without any chaotic behavior. Furthermore, we extend the framework of QSLs to operator flows \cite{Carabba22,Hornedal23}. In the case of quantum state evolution, two milestones are the celebrated results by Mandelstamm and Tamm (MT) \cite{Mandelstam45} and by Margolus and Levitin (ML) \cite{Margolus98}. We derive two generalized MT and ML-type of operator QSLs (OQSLs) and assess the existence of a crossover between them, which we illustrate with a qubit and a random matrix Hamiltonian. We further apply our results to the time evolution of autocorrelation functions, obtaining computable constraints on the linear dynamical response of quantum systems out of equilibrium and the quantum Fisher information governing the precision in quantum parameter estimation. Finally, using a geometric approach, we introduce an additional OQSL and show the conditions for its saturation, that is, for the dynamics to be geodesic. This result holds for operator flows induced by arbitrary unitaries, i.e., with time- or parameter-dependent generators. The geometric OQSL is applied to the Wegner flow equations in Hamiltonian renormalization group theory and to the Krylov complexity quantifying operator growth. We finally show the equivalence between the saturation of the dispersion bound and that of the geometric OQSL. In other words, complexity grows at the maximal rate when it follows a geodesic trajectory.