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Quantum algorithms for simulating Hamiltonian dynamics have been extensively developed, but there has been much less work on quantum algorithms for simulating the dynamics of open quantum systems.
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Frank Verstraete, Michael M. Wolf, and Juan Ignacio Cirac, Quantum computation and quantum-state engineering driven by dissipation , Nature Physics 5
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Chaobin Liu and Radhakrishnan Balu, Continuous-time open quantum walks , 2016, arXiv:1604.05652
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