Krylov method and its’ application to Quantum Control

Over the last few years, Krylov (complexity) methods (i.e. using tridiagonal matrices) have been extensively employed to study dynamical properties of many-body quantum systems. I will briefly review the formalism and describe the operator growth hypothesis [1]. Remarkably, the Krylov method is also useful for describing the Adiabatic Gauge Potential (AGP), which is the generator of adiabatic deformations [2]. Having a complete understanding of the AGP operator enables us to perform quantum control. Using the Krylov method method, I will describe a formalism for computing the AGP operator. I will discuss the advantage of using this method. I will then present a systematic procedure for approximating the AGP and discuss this approximation in different types of quantum systems. Finally, I will compare the quantum-chaos-probing capabilities of the norm of the AGP operator and Krylov complexity [1] and infer some plausible interesting physics with regards to quantum chaotic dynamics. I will end with some open questions and future directions.