Extremal numbers for cycles in a hypercube

Let $ex(Q_n,H)$ be the largest number of edges in a subgraph $G$ of a hypercube $Q_n$ such that there is no subgraph of $G$ isomorphic to $H$. We show that for any integer $k\ge 3$, $$ex(Q_n,C_{4k+2})=O(n^{\frac{5}{6}+\frac{1}{3(2k-2)}}{2}^n)\mathrm{.}$$