  We investigate the excitation dynamics at a first-order quantum phase
transition (QPT). More specifically, we consider the quench-induced QPT in the
quantum search algorithm, which aims at finding out a marked element in an
unstructured list. We begin by deriving the exact dynamics of the model, which
is shown to obey a Riccati differential equation. Then, we discuss the
probabilities of success by adopting either global or local adiabaticity
strategies. Moreover, we determine the disturbance of the quantum criticality
as a function of the system size. In particular, we show that the critical
point exponentially converges to its thermodynamic limit even in a fast
evolution regime, which is characterized by both entanglement QPT estimators
and the Schmidt gap. The excitation pattern is manifested in terms of quantum
domains walls separated by kinks. The kink density is then shown to follow an
exponential scaling as a function of the evolution speed, which can be
interpreted as a Kibble-Zurek mechanism for first-order QPTs.
