Here, we use a symmetric protocol consisting of T steps, where the perturbation and propagation steps are alternated with either a or K appearing at both the beginning and the end, so that ⇤ = (a 1 , K 1 , a 2 , K 2 , . . . , K T , a T +1 ) = ⇤, where ⇤ is the reverse protocol. This protocol produces a trajectory X ⌘ (x 0 , x 1 , . . . , x T ). To generate the appropriate acceptance for an NCMC move to maintain detailed balance, we also have to consider the probability of observing a time-reversed trajectory