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Density Matrix Exponentiation is a technique for simulating Hamiltoniandynamics when the Hamiltonian to be simulated is available as a quantum state.In this paper, we present a natural analogue to this technique, for simulatingMarkovian dynamics governed by the well known Lindblad master equation. Forthis purpose, we first propose an input model in which a Lindblad operator $L$is encoded into a quantum state $\psi$. Then, given access to $n$ copies of thestate $\psi$, the task is to simulate the corresponding Markovian dynamics fortime $t$. We propose a quantum algorithm for this task, called Wave MatrixLindbladization, and we also investigate its sample complexity. We show thatour algorithm uses $n = O(t^2/\varepsilon)$ samples of $\psi$ to achieve thetarget dynamics, with an approximation error of $O(\varepsilon)$.

Wave Matrix Lindbladization I: Quantum Programs for Simulating Markovian Dynamics