Generalized Krylov Complexity

TL;DR

This paper generalizes Krylov complexity to include multiple generators and continuous symmetries, introducing a new framework and algorithm for measuring state complexity in quantum systems with diverse transformations.

Contribution

It develops a generalized Krylov complexity framework for multi-generator unitary evolutions and introduces a novel orthogonalization algorithm structured as a network of orthogonal blocks.

Findings

Framework applicable to continuous symmetries

New orthogonalization algorithm for complexity measurement

Explicit examples demonstrating practical application

Abstract

We extend the concept of Krylov complexity to include 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 where different directions of transformation lead to varying degrees of complexity, which can be compared to geometric approaches to understanding complexity, such as Nielsen complexity. In this context, we introduce a generalized orthogonalization algorithm and delineate its computational framework, which is structured as a network of orthogonal blocks rather than a simple linear chain. Additionally, we provide explicit evaluations of specific illustrative examples to demonstrate the practical application of this framework.