Generalized Krylov complexity

We extend the concept of Krylov complexity to encompass general unitary evolutions involving multiple generators. This generalization enables us to formulate a framework for generalized Krylov complexity, which serves as a measure of the complexity of states associated with continuous symmetries within a model. Furthermore, we investigate scenarios in which different directions of transformation lead to varying degrees of complexity, in close analogy with geometric approaches to complexity, such as Nielsen’s framework. In this context, we introduce a generalized orthogonalization algorithm and delineate its computational structure, which is organized as a network of orthogonal blocks rather than a simple linear chain. Finally, we present explicit evaluations of several illustrative examples to demonstrate the practical application of this framework.