  In this article, we consider the preconditioned Hamiltonian Monte Carlo
(pHMC) algorithm defined directly on an infinite-dimensional Hilbert space. In
this context, and under a condition reminiscent of strong log-concavity of the
target measure, we prove convergence bounds for adjusted pHMC in the standard
1-Wasserstein distance. The arguments rely on a synchronous coupling of two
copies of pHMC, which is controlled by adapting elements from arXiv:1805.00452.
