  We study the operator growth problem and its complexity in the many-body
localization (MBL) system from the Lanczos algorithm perspective. Using the
Krylov basis, the operator growth problem can be viewed as a single-particle
hopping problem on a semi-infinite chain with the hopping amplitudes given by
the Lanczos coefficients. We find that, in the MBL systems, the Lanczos
coefficients scale as $\sim n/\ln(n)$ asymptotically, same as in the ergodic
systems, but with an additional even-odd alteration and an effective
randomness. We use a simple linear extrapolation scheme as an attempt to
extrapolate the Lanczos coefficients to the thermodynamic limit. With the
original and extrapolated Lanczos coefficients, we study the properties of the
emergent single-particle hopping problem via its spectral function, integrals
of motion, Krylov complexity, wavefunction profile and return probability. Our
numerical results of the above quantities suggest that the emergent
single-particle hopping problem in the MBL system is localized when initialized
on the first site. We also study the operator growth in the MBL
phenomenological model, whose Lanczos coefficients also have an even-odd
alteration, but approach constants asymptotically. The Krylov complexity grows
linearly in time in this case.

Disordered Systems and Neural Networks;Quantum Physics