A novel unadjusted Hamiltonian Monte Carlo (uHMC) algorithm is suggested that uses a stratified Monte Carlo (SMC) time integrator for the underlying Hamiltonian dynamics in place of the usual Verlet time integrator. For target distributions of the form µ(dx) ∝ e −U (x) dx where U ∶ R d → R ≥0 is both K-strongly convex and L-gradient Lipschitz, and initial distributions ν with finite second moment, coupling proofs reveal that an ε-accurate approximation of the target distribution µ in L 2 -Wasserstein distance W 2 can be achieved by the uHMC algorithm with SMC time integration using O (d K) 1 3 (L K) 5 3 ε −2 3 log(W 2 (µ, ν) ε) + gradient evaluations; whereas without any additional assumptions the corresponding complexity of the uHMC algorithm with Verlet time integration is in general O (d K) 1 2 (L K) 2 ε −1 log(W 2 (µ, ν) ε) + . The SMC time integrator involves a minor modification to Verlet, and hence, is easy to implement.