A quantum algorithm is presented for the simulation of arbitrary Markovian dynamics of a qubit, described by a semigroup of single qubit quantum channels ${T_t}$ specified by a generator $mathcal{L}$. This algorithm requires only $mathcal{O}big((||mathcal{L}||_{(1rightarrow 1)} t)^{3/2}/epsilon^{1/2} big)$ single qubit and CNOT gates and approximates the channel $T_t = e^{tmathcal{L}}$ up to chosen accuracy $epsilon$. Inspired by developments in Hamiltonian simulation, a decomposition and recombination technique is utilised which allows for the exploitation of recently developed methods for the approximation of arbitrary single-qubit channels. In particular, as a result of these methods the algorithm requires only a single ancilla qubit, the minimal possible dilation for a non-unitary single-qubit quantum channel.
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