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First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. One of the most common reasons for using Markov chain Monte Carlo (MCMC) is to estimate the value of an otherwise intractable integral. Typically MCMC algorithms will give an exact answer as the number of iterations increases to infinity. However, this gives little assurance about the precision of the estimate in finite samples. This paper addresses this important issue from a theoretical point of view for the Hamiltonian Monte Carlo (HMC) algorithm, an algorithm which has been receiving a fair amount of recent attention.