Since precisely controlling dissipation in realistic environments is challenging, digital simulation of the Lindblad master equation (LME) is of great significance for understanding nonequilibrium dynamics in open quantum systems. However, achieving long-time simulations for complex systems with multiple dissipation channels remains a major challenge, both theoretically and experimentally. Here, we propose a 1-dilation digital scheme for simulating the LME based on quantum trajectory averaging without postselection. By rigorously matching the stochasticity inherent in quantum trajectories with the probabilistic outcomes of quantum measurements, our method effectively translates the classically established quantum jump algorithm into executable quantum circuits. A key advantage of our method is that it overcomes the exponential suppression of success probability seen in some existing postselection-dependent schemes, especially for long-time evolution or systems with numerous jump operators. Moreover, the scheme can be extended to a 2-dilation framework for the nonlinear LME with postselection, bridging the full LME and non-Hermitian Hamiltonian dynamics. This extended scheme provides a digital approach for exploring the interplay between non-Hermitian Hamiltonians and dissipative terms within a monitored quantum dynamics framework.

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• Localization
• Open quantum systems
• Quantum algorithms & computation
• Quantum circuits
• Quantum simulation
• Skin effect