  We study random quantum circuits and their rate of producing bipartite
entanglement, specifically with respect to the choice of 2-qubit gates and the
order (protocol) in which these are applied. The problem is mapped to a
Markovian process and proved that there are large spectral equivalence classes
-- different configurations have the same spectrum. Optimal gates and the
protocol that generate entanglement with the fastest theoretically possible
rate are identified. Relaxation towards the asymptotic thermal entanglement
proceeds via a series of phase transitions in the local relaxation rate, which
is a consequence of non-Hermiticity. In particular, non-Hermiticity can cause
the rate to be either faster, or, even more interestingly, slower than
predicted by the matrix eigenvalue gap. This is caused by an exponential in
system size explosion of expansion coefficient sizes resulting in a 'phantom'
eigenvalue, and is due to non-orthogonality of non-Hermitian eigenvectors. We
numerically demonstrate that the phenomenon occurs also in random circuits with
non-optimal generic gates, random U(4) gates, and also without spatial or
temporal randomness, suggesting that it could be of wide importance also in
other non-Hermitian settings, including correlations.
