  Over the past decade classical optical systems with gain or loss, modelled by
non-Hermitian parity-time symmetric Hamiltonians, have been deeply
investigated. Yet, their applicability to the quantum domain with
number-resolved photonic states is fundamentally voided by quantum-limited
amplifier noise. Here, we show that second-quantised Hermitian Hamiltonians on
the Fock space give rise to non-Hermitian effective Hamiltonians that generate
the dynamics of corresponding creation and annihilation operators. Using this
equivalence between $\mathcal{PT}$-symmetry and symplectic Bogoliubov
transformations, we create a quantum optical scheme comprising squeezing,
phase-shifters, and beam-splitters for simulating arbitrary non-unitary
processes by way of singular value decomposition. In contrast to the
post-selection scheme for non-Hermitian quantum simulation, the success
probability in this approach is independent of the system size or simulation
time, and can be efficiently Trotterised similar to a unitary transformation.
