  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. Proposals to realize
high-order EPs have been based on the use of non-Hermitian Hamiltonians (NHHs)
of composite systems, i.e., the operators describing the evolution of coupled
post-selected systems or coupled intense light fields. In both cases, quantum
jumps play no role. 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 the new NHHs describing,
e.g., networks of coupled resonators. Notably, such a mapping allows to
correctly reproduce the results of the Liouvillian dynamics, including quantum
jumps. By applying this mapping, we demonstrate that quantum EPs of any order
can be engineered in dissipative systems and can, thus, be probed by the
coherence and spectral functions. As an example, we consider a $U(1)$-symmetric
quadratic Liouvillian describing an optical cavity with incoherent mode
coupling, which can also possess anti-$\cal PT$-symmetry. Compared to their
$\cal PT$-symmetric counterparts, such anti-$\cal PT$-symmetric systems could
be easier to scale and, thus, can serve as a promising platform for engineering
quantum systems with high-order EPs.
