  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.
