  We investigate the quantum reaction-diffusion dynamics of fermionic particles
which coherently hop in a one-dimensional lattice and undergo annihilation
reactions. The latter are modelled as dissipative processes which involve
losses of pairs $2A \to \emptyset$, triplets $3A \to \emptyset$, and
quadruplets $4A \to \emptyset$ of neighbouring particles. When considering
classical particles, the corresponding decay of their density in time follows
an asymptotic power-law behavior. The associated exponent in one dimension is
different from the mean-field prediction whenever diffusive mixing is not too
strong and spatial correlations are relevant. This specifically applies to
$2A\to \emptyset$, while the mean-field power-law prediction just acquires a
logarithmic correction for $3A \to \emptyset$ and is exact for $4A \to
\emptyset$. A mean-field approach is also valid, for all the three processes,
when the diffusive mixing is strong, i.e., in the so-called reaction-limited
regime. Here, we show that the picture is different for quantum systems. We
consider the quantum reaction-limited regime and we show that for all the three
processes power-law behavior beyond mean field is present as a consequence of
quantum coherences, which are not related to space dimensionality. The decay in
$3A\to \emptyset$ is further, highly intricate, since the power-law behavior
therein only appears within an intermediate time window, while at long times
the density decay is not power-law. Our results show that emergent critical
behavior in quantum dynamics has a markedly different origin, based on quantum
coherences, to that applying to classical critical phenomena, which is,
instead, solely determined by the relevance of spatial correlations.
