  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
$\mathcal{O}\big((||\mathcal{L}||_{(1\rightarrow 1)} t)^{3/2}/\epsilon^{1/2}
\big)$ single qubit and CNOT gates and approximates the channel $T_t =
e^{t\mathcal{L}}$ 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.
