  The hypercontractivity inequality for the qubit depolarizing channel $\Psi_t$
states that $\|\Psi_t^{\otimes n}(X)\|_p\leq \|X\|_q$ provided that $p\geq q>
1$ and $t\geq \ln \sqrt{\frac{p-1}{q-1}}$. In this paper we present an
improvement of this inequality. We first prove an improved quantum
logarithmic-Sobolev inequality and then use the well-known equivalence of
logarithmic-Sobolev inequalities and hypercontractivity inequalities to obtain
our main result. As applications of these results, we present an asymptotically
tight quantum Faber-Krahn inequality on the hypercube, and a new quantum
Schwartz-Zippel lemma.
