  We show that in a quantum system evolving unitarily under a stochastic
quantum circuit the notions of irreversibility, universality of computation,
and entanglement are closely related. As the state evolves from an initial
product state, it gets asymptotically maximally entangled. We define
irreversibility as the failure of searching for a disentangling circuit using a
Metropolis-like algorithm. We show that irreversibility corresponds to
Wigner-Dyson statistics in the level spacing of the entanglement eigenvalues,
and that this is obtained from a quantum circuit made from a set of universal
gates for quantum computation. If, on the other hand, the system is evolved
with a non-universal set of gates, the statistics of the entanglement level
spacing deviates from Wigner-Dyson and the disentangling algorithm succeeds.
These results open a new way to characterize irreversibility in quantum
systems.
