Summary: We establish \(L^{\mathfrak{q}}\) convergence for Hamiltonian Monte Carlo (HMC) algorithms. More specifically, under mild conditions for the associated Hamiltonian motion, we show that the outputs of the algorithms converge (strongly for \(2\leq\mathfrak{q}<\infty\) and weakly for \(1<\mathfrak{q}<2\)) to the desired target distribution. In addition, we establish a general convergence rate for an \(L^{\mathfrak{q}}\) convergence given a convergence rate at a specific \(q^*\), and apply this result to conclude geometric convergence in the Euclidean space for HMC with uniformly strongly logarithmic concave target and auxiliary distributions. We also present the results of experiments to illustrate convergence in \(L^{\mathfrak{q}} \).