  Non-unitary state preparation protocols such as imaginary time evolution
(ITE) offer substantial advantages relative to unitary ones, including the
ability to prepare certain long-range correlated states more efficiently. Here,
we ask whether such protocols are also robust to noise arising due to coupling
to the environment. We consider a non-unitary ITE "circuit" subjected to a
variety of noise models and investigate whether the resulting steady state
remains in the same phase as the target state of the ITE at finite noise
strength. Taking the one-dimensional quantum Ising model as a concrete example,
we find that the ground state order and associated phase transition persist in
the presence of noise, provided the noise does not explicitly break the
symmetry that protects the phase transition. That is, the noise must possess
the protecting symmetry in a weak (or average) form. Our analysis is
facilitated by a mapping to an effective Hamiltonian picture in a doubled
Hilbert space. We discuss possible implications of these findings for quantum
simulation on noisy quantum hardware.
