  We give lower bounds on the performance of two of the most popular sampling
methods in practice, the Metropolis-adjusted Langevin algorithm (MALA) and
multi-step Hamiltonian Monte Carlo (HMC) with a leapfrog integrator, when
applied to well-conditioned distributions. Our main result is a nearly-tight
lower bound of $\widetilde{\Omega}(\kappa d)$ on the mixing time of MALA from
an exponentially warm start, matching a line of algorithmic results up to
logarithmic factors and answering an open question of Chewi et. al. We also
show that a polynomial dependence on dimension is necessary for the relaxation
time of HMC under any number of leapfrog steps, and bound the gains achievable
by changing the step count. Our HMC analysis draws upon a novel connection
between leapfrog integration and Chebyshev polynomials, which may be of
independent interest.
