  Recently, there has been intense research in proposing and developing various
methods for constructing high-order exceptional points (EPs) in dissipative
systems. These EPs can possess a number of intriguing properties related to,
e.g., chiral transport and enhanced sensitivity. Previous proposals to realize
non-Hermitian Hamiltonians (NHHs) with high-order EPs have been mainly based on
either direct construction of spatial networks of coupled modes or utilization
of synthetic dimensions, e.g., of mapping spatial lattices to time or
photon-number space. Both methods rely on the construction of effective NHHs
describing classical or postselected quantum fields, which neglect the effects
of quantum jumps, and which, thus, suffer from a scalability problem in the
{\it quantum regime}, when the probability of quantum jumps increases with the
number of excitations and dissipation rate. Here, by considering the full
quantum dynamics of a quadratic Liouvillian superoperator, we introduce a
simple and effective method for engineering NHHs with high-order quantum EPs,
derived from evolution matrices of system operators moments. That is, by
quantizing higher-order moments of system operators, e.g., of a quadratic
two-mode system, the resulting evolution matrices can be interpreted as
alternative NHHs describing, e.g., a spatial lattice of coupled resonators,
where spatial sites are represented by high-order field moments in the
synthetic space of field moments. As an example, we consider a $U(1)$-symmetric
quadratic Liouvillian describing a {\it bimodal} cavity with incoherent mode
coupling, which can also possess anti-$\cal PT$-symmetry, whose field moment
dynamics can be mapped to an NHH governing a spatial {\it network} of coupled
resonators with high-order EPs.
