  We describe a quantum algorithm to estimate the $\alpha$-Renyi entropy of an
unknown density matrix $\rho\in\mathcal{C}^{d\times d}$ for $\alpha\neq 1$ by
combining the recent technique of quantum singular value transformations with
the method of estimating normalised traces in the one clean qubit model. We
consider an oracular input model where the input state is prepared via a
quantum oracle that outputs a purified version of the state, assumed to be
non-singular. Our method outputs an estimate of the $\alpha$-Renyi entropy to
additive precision $\epsilon$, using an expected total number
$O\left(\frac{1}{(x\epsilon)^2}\right)$ of independent applications of a
quantum circuit which coherently queries the input unitary
$O\left(\frac{1}{\delta}\log \frac{d}{\epsilon}\right)$ times, in each case
measuring a single output qubit. Here $\delta$ is a lower cutoff on the
smallest eigenvalue of $\rho$ and $x=\frac{1}{d}\!\mathop{Tr}{\rho^\alpha}$.
The expected number of measurements made in this method can be compared to
results in the sample complexity model that generally require
$\Theta(d^2/\epsilon^2)$ samples. Furthermore, we also show that multiplicative
approximations can be obtained by iteratively using additive approximations,
with an overhead logarithmic in the dimension $d$.
