One of the most promising applications of quantum computing is the simulation of quantum systems. The goal is to construct a quantum algorithm that closely approximates the solution to Schrödinger’s equation, which is a unitary propagator in time. Much attention has been given to this problem, and modern approaches provide a variety of highly efficient algorithms. The lesser-studied Lindblad equation generalizes the Schrödinger equation to quantum systems that undergo dissipation, leading to non-unitary dynamics that prevent a naïve application of state-of-the-art quantum algorithms. In this work, we utilize a correspondence between repeated interaction CPTP maps and Lindbladian dynamics to formulate an embedding of the non-unitary dynamics in a higher dimensional space that evolves under a Hamiltonian with low space overhead, which we can simulate with efficient quantum algorithms. In the process, we derive error bounds on the approximate correspondence and provide bounds on the computational complexity of the approach.