  Characterizing the quantum complexity of local random quantum circuits is a
very deep problem with implications to the seemingly disparate fields of
quantum information theory, quantum many-body physics and high energy physics.
While our theoretical understanding of these systems has progressed in recent
years, numerical approaches for studying these models remains severely limited.
In this paper, we discuss a special class of numerically tractable quantum
circuits, known as quantum automaton circuits, which may be particularly well
suited for this task. These are circuits which preserve the computational
basis, yet can produce highly entangled output wave functions. Using ideas from
quantum complexity theory, especially those concerning unitary designs, we
argue that automaton wave functions have high quantum state complexity. We look
at a wide variety of metrics, including measurements of the output bit-string
distribution and characterization of the generalized entanglement properties of
the quantum state, and find that automaton wave functions closely approximate
the behavior of fully Haar random states. In addition to this, we identify the
generalized out-of-time ordered 2k-point correlation functions as a
particularly useful probe of complexity in automaton circuits. Using these
correlators, we are able to numerically study the growth of complexity well
beyond the scrambling time for very large systems. As a result, we are able to
present evidence of a linear growth of design complexity in local quantum
circuits, consistent with conjectures from quantum information theory.
