Dissipation has been recently demonstrated as a powerful primitive for designing quantum algorithms. We apply this viewpoint directly to linear-system solving, Ax=b. We construct a simple purely dissipative Lindbladian whose unique fixed point encod
Dissipation has been recently demonstrated as a powerful primitive for designing quantum algorithms. We apply this viewpoint directly to linear-system solving, Ax=b. We construct a simple purely dissipative Lindbladian whose unique fixed point encodes the linear-system solution. We prove dimension-independent trace-distance mixing in \Theta(\kappa^2\log(1/\eps)) time. We then show how to run this Lindbladian on digital quantum computers via collective block encoding and Lindbladian simulation, resulting in an $O\!\left(\kappa^2\log(1/\eps) \frac{\log(\kappa/\eps)}{\log\log(\kappa/\eps)}\right) query complexity for both U_A, the block encoding of A, and U_b, the |b\rangle$ state preparation unitary.