  Quantum computers have long been anticipated to excel in simulating quantum
many-body physics. While most previous work has focused on Hermitian physics,
we demonstrate the power of variational quantum circuits for resource-efficient
simulations of dynamical and equilibrium physics in non-Hermitian systems,
revealing new phenomena beyond standard Hermitian quantum machines. Using a
variational quantum compilation scheme for fermionic systems, we reduce gate
count, save qubits, and eliminate the need for postselection, a major challenge
in simulating non-Hermitian dynamics via standard Trotterization.
Experimentally, we observed a supersonic mode in the connected density-density
correlation function on an $ n = 18 $ fermionic chain after a non-Hermitian,
locally interacting quench, which would otherwise be forbidden by the
Lieb-Robinson bound in a Hermitian system. Additionally, we investigate
sequential quantum circuits generated by tensor networks for ground state
preparation, here defined as the eigenstate with the lowest real part
eigenvalue, using a variance minimization scheme. Through a trapped-ion
implementation on the Quantinuum H1 quantum processor, we accurately capture
correlation functions and energies across an exceptional point on a dissipative
spin chain up to length $ n = 20 $ using only 3 qubits. Motivated by these
advancements, we provide an analytical example demonstrating that simulating
single-qubit non-Hermitian dynamics for $\Theta(\log(n))$ time from certain
initial states is exponentially hard on a quantum computer, offering insights
into the opportunities and limitations of using quantum computation for
simulating non-Hermitian physics.
