Motivated by the recent progress of quantum chaos and quantum information scrambling, the growth of an operator under the Heisenberg evolution has attracted a lot of attentions. I will first introduce a recently proposed perspective on the operator growth problem from the Lanczos algorithm point of view and the associated “Krylov complexity”. While reviewing some of the recent results of Krylov complexity of the chaotic systems and integrable systems by other researchers, I will touch on our results about the  “Krylov complexity” of the many-body localization (MBL) systems. In particular, we find strong numerical evidences that the Krylov complexity of MBL is bounded in time, suggesting that the Lanczos algorithm could be an efficient way in simulating the operator dynamics in the MBL systems. I will also mention some very recent works by other people on generalizing ``Krylov “complexity” to ``spread complexity” and to open systems, and some potential future directions.

(Pizza and refreshments will be served after the talk.)