  Certain aspects of some unitary quantum systems are well-described by
evolution via a non-Hermitian effective Hamiltonian, as in the Wigner-Weisskopf
theory for spontaneous decay. Conversely, any non-Hermitian Hamiltonian
evolution can be accommodated in a corresponding unitary system + environment
model via a generalization of Wigner-Weisskopf theory. This demonstrates the
physical relevance of novel features such as exceptional points in quantum
dynamics, and opens up avenues for studying many body systems in the complex
plane of coupling constants. In the case of lattice field theory, sparsity
lends these channels the promise of efficient simulation on standardized
quantum hardware. We thus consider quantum operations that correspond to
Suzuki-Lee-Trotter approximation of lattice field theories undergoing
non-Hermitian time evolution, with potential applicability to studies of spin
or gauge models at finite chemical potential, with topological terms, to
quantum phase transitions - a range of models with sign problems. We develop
non-Hermitian quantum circuits and explore their promise on a benchmark, the
quantum one-dimensional Ising model with complex longitudinal magnetic field,
showing that observables can probe the Lee-Yang edge singularity. The
development of attractors past critical points in the space of complex
couplings indicates a potential for study on near-term noisy hardware.
