I used this paper more to get intuition on HMC rather than to think about their research contribution, but it might be useful for me to dive into this later, about how they can eliminate the rejection stage (which happens only due to computer implementation). They have an interesting presentation based on moving across a ladder and jumping among the two sides.
This representation of the actions performed in HMC, and the corresponding state space, is unique to this paper and diverges from the typical presentation of HMC.