The implementation and practicality of quantum algorithms hinge largely on the quality of operations within a quantum processor. Therefore, including realistic error models in quantum computing simulation platforms is crucial for testing these algorithms. Existing classical simulation techniques of quantum information processing devices exhibit a tradeoff between scalability (the number of qubits that can be simulated) and accuracy (how close the simulation is to the target error model). In this paper, we introduce a simulation approach that relies on approximating the density matrix evolution with a stochastic sum of unitary and measurement channels within a pure-state simulation environment. This model shows an improvement of at least one order of magnitude in terms of accuracy compared to the best known stochastic approaches while allowing us to simulate a larger number of qubits compared to the exact density matrix simulation. Furthermore, we used this approach to realistically simulate Grover's algorithm and the surface code 17 using a gate set tomography characterization of quantum operations as a noise model.