  We consider the Nielsen complexity ${\mathcal C}_N$, the Loschmidt echo
${\mathcal L}$, and the Fubini-Study complexity $\tau$ in the transverse XY
model, following a sudden quantum quench, in the thermodynamic limit. At small
times, the first two are related by ${\mathcal L} \sim e^{-{\mathcal C}_N}$. By
computing a novel time-dependent quantum information metric, we show that in
this regime, ${\mathcal C}_N \sim d\tau^2$, up to lowest order in perturbation.
The former relation continues to hold in the same limit at large times, whereas
the latter does not. Our results indicate that in the thermodynamic limit, the
Nielsen complexity and the Loschmidt echo show enhanced temporal oscillations
when one quenches from a close neighbourhood of the critical line, while such
oscillations are notably absent when the quench is on such a line. We explain
this behaviour by studying the nature of quasi-particle excitations in the
vicinity of criticality. Finally, we argue that the triangle inequality for the
Nielsen complexity might be violated in certain regions of the parameter space,
and point out why one should be careful about the nature of the interaction
Hamiltonian, while using this measure.
