  In this paper, we introduce the $\ell_p^p$-error metric (for $p \geq 2$) when
answering linear queries under the constraint of differential privacy. We
characterize such an error under $(\epsilon,\delta)$-differential privacy.
Before this paper, tight characterization in the hardness of privately
answering linear queries was known under $\ell_2^2$-error metric (Edmonds et
al., STOC 2020) and $\ell_p^2$-error metric for unbiased mechanisms (Nikolov
and Tang, ITCS 2024). As a direct consequence of our results, we give tight
bounds on answering prefix sum and parity queries under differential privacy
for all constant $p$ in terms of the $\ell_p^p$ error, generalizing the bounds
in Henzinger et al. (SODA 2023) for $p=2$.
