  Quantum speed limits set an upper bound to the rate at which a quantum system
can evolve and as such can be used to analyze the scrambling of information. To
this end, we consider the survival probability of a thermofield double state
under unitary time-evolution which is related to the analytic continuation of
the partition function. We provide an exponential lower bound to the survival
probability with a rate governed by the inverse of the energy fluctuations of
the initial state. Further, we elucidate universal features of the
non-exponential behavior at short and long times of evolution that follow from
the analytic properties of the survival probability and its Fourier transform,
both for systems with a continuous and a discrete energy spectrum. We find the
spectral form factor in a number of illustrative models, notably we obtain the
exact answer in the Gaussian unitary ensemble for any $N$ with excellent
agreement with recent numerical studies. We also discuss the relationship of
our findings to models of black hole information loss, such as the
Sachdev-Ye-Kitaev model dual to AdS$_2$ as well as higher-dimensional versions
of AdS/CFT.
