  We study the operator growth in open quantum systems with dephasing
dissipation terms, extending the Krylov complexity formalism of Phys. Rev. X 9,
041017. Our results are based on the study of the dissipative $q$-body
Sachdev-Ye-Kitaev (SYK$_q$) model, governed by the Markovian dynamics. We
introduce a notion of ''operator size concentration'' which allows a
diagrammatic and combinatorial proof of the asymptotic linear behavior of the
two sets of Lanczos coefficients ($a_n$ and $b_n$) in the large $q$ limit. Our
results corroborate with the semi-analytics in finite $q$ in the large $N$
limit, and the numerical Arnoldi iteration in finite $q$ and finite $N$ limit.
As a result, Krylov complexity exhibits exponential growth following a
saturation at a time that grows logarithmically with the inverse dissipation
strength. The growth of complexity is suppressed compared to the closed system
results, yet it upper bounds the growth of the normalized out-of-time-ordered
correlator (OTOC). We provide a plausible explanation of the results from the
dual gravitational side.
