  Simulating the dynamics and the non-equilibrium steady state of an open
quantum system are hard computational tasks on conventional computers. For the
simulation of the time evolution, several efficient quantum algorithms have
recently been developed. However, computing the non-equilibrium steady state as
the long-time limit of the system dynamics is often not a viable solution,
because of exceedingly long transient features or strong quantum correlations
in the dynamics. Here, we develop an efficient quantum algorithm for the direct
estimation of averaged expectation values of observables on the non-equilibrium
steady state, thus bypassing the time integration of the master equation. The
algorithm encodes the vectorized representation of the density matrix on a
quantum register, and makes use of quantum phase estimation to approximate the
eigenvector associated to the zero eigenvalue of the generator of the system
dynamics. We show that the output state of the algorithm allows to estimate
expectation values of observables on the steady state. Away from critical
points, where the Liouvillian gap scales as a power law of the system size, the
quantum algorithm performs with exponential advantage compared to exact
diagonalization.
