  Recently, Akers et al. proposed a non-isometric holographic map from the
interior of a black hole to its exterior. Within this model, we study
properties of the black hole $S$-matrix, which are in principle accessible to
observers who stay outside the black hole. Specifically, we investigate a
scenario in which an infalling agent interacts with radiation both outside and
inside the black hole. Because the holographic map involves postselection, the
unitarity of the $S$-matrix is not guaranteed in this scenario, but we find
that unitarity is satisfied to very high precision if suitable conditions are
met. If the internal black hole dynamics is described by a pseudorandom unitary
transformation, and if the operations performed by the infaller have
computational complexity scaling polynomially with the black hole entropy, then
the $S$-matrix is unitary up to corrections that are superpolynomially small in
the black hole entropy. Furthermore, while in principle quantum computation
assisted by postselection can be very powerful, we find under similar
assumptions that the $S$-matrix of an evaporating black hole has polynomial
computational complexity.
