On Estimating Quantum Divergences with $p$-Power Means

arXiv:2611.01163v1 Announce Type: new Abstract: We study the problem of estimating the quantum divergence with $p$-power means, $\Phi_p(\rho\,\|\,\sigma)$. This quantity was investigated by Lam and Le (\textit{Linear Algebra Appl.}\ 2021). For any constant $p \in (0, 1)$, we present sample-efficient and query-efficient quantum algorithms for estimating $\Phi_p(\rho\,\|\,\sigma)$ and further show that estimating $\Phi_p(\rho\,\|\,\sigma)$ is $\mathsf{BQP}$-complete in the low-rank case.