  Non-equilibrium steady states are a focal point of research in the study of
open quantum systems. Previous variational algorithms for searching these
steady states have suffered from resource-intensive implementations due to
vectorization or purification of the system density matrix, requiring large
qubit resources and long-range coupling. In this work, we present a novel
variational quantum algorithm that efficiently searches for non-equilibrium
steady states by simulating the operator-sum form of the Lindblad equation. By
introducing the technique of random measurement, we are able to estimate the
nonlinear cost function while reducing the required qubit resources by half
compared to previous methods. Additionally, we prove the existence of the
parameter shift rule in our variational algorithm, enabling efficient updates
of circuit parameters using gradient-based classical algorithms. To demonstrate
the performance of our algorithm, we conduct simulations for dissipative
quantum transverse Ising and Heisenberg models, achieving highly accurate
results. Our approach offers a promising solution for effectively addressing
non-equilibrium steady state problems while overcoming computational
limitations and implementation challenges.
