  The gradient descent approach is the key ingredient in variational quantum
algorithms and machine learning tasks, which is an optimization algorithm for
finding a local minimum of an objective function. The quantum versions of
gradient descent have been investigated and implemented in calculating
molecular ground states and optimizing polynomial functions. Based on the
quantum gradient descent algorithm and Choi-Jamiolkowski isomorphism, we
present approaches to simulate efficiently the nonequilibrium steady states of
Markovian open quantum many-body systems. Two strategies are developed to
evaluate the expectation values of physical observables on the nonequilibrium
steady states. Moreover, we adapt the quantum gradient descent algorithm to
solve linear algebra problems including linear systems of equations and
matrix-vector multiplications, by converting these algebraic problems into the
simulations of closed quantum systems with well-defined Hamiltonians. Detailed
examples are given to test numerically the effectiveness of the proposed
algorithms for the dissipative quantum transverse Ising models and
matrix-vector multiplications.
