Krylov Complexity of Fermionic and Bosonic Gaussian States

The concept of \emph{complexity} has become pivotal in multiple disciplines, including quantum information, where it serves as an alternative metric for gauging the chaotic evolution of a quantum state. This paper focuses on \emph{Krylov complexity}, a specialized form of quantum complexity that offers an unambiguous and intrinsically meaningful assessment of the spread of a quantum state over all possible orthogonal bases. Our study is situated in the context of Gaussian quantum states, which are fundamental to both Bosonic and Fermionic systems and can be fully described by a covariance matrix. We show that while the covariance matrix is essential, it is insufficient alone for calculating Krylov complexity due to its lack of relative phase information. Our findings suggest that the relative covariance matrix can provide an upper bound for Krylov complexity for Gaussian quantum states. We also explore the implications of Krylov complexity for theories proposing complexity as a candidate for holographic duality by computing Krylov complexity for the thermofield double States (TFD) and Dirac field.