  We demonstrate a relation between Nielsen's approach towards circuit
complexity and Krylov complexity through a particular construction of quantum
state space geometry. We start by associating K\"ahler structures on the full
projective Hilbert space of low rank algebras. This geometric structure of the
states in the Hilbert space ensures that every unitary transformation of the
associated algebras leave the metric and the symplectic forms invariant. We
further associate a classical matter free Jackiw-Teitelboim (JT) gravity model
with these state manifolds and show that the dilaton can be interpreted as the
quantum mechanical expectation values of the symmetry generators. On the other
hand we identify the dilaton with the spread complexity over a Krylov basis
thereby proposing a geometric perspective connecting two different notions of
complexity.
