  We study a simple quantum circuit model which, without need for fine tuning,
is very close to sitting at the transition between ergodic and many-body
localized (MBL) phases. We probe the properties of the model on large
finite-size systems, using a matrix-free exact diagonalization method that
takes advantage of the shallow nature of the circuit. Moreover, we provide a
qualitative entanglement bottleneck picture to account for the
close-to-critical nature of the model.
