We argue that the Anti-de-Sitter (AdS) geometry in $d+1$ dimensions
naturally emerges from an arbitrary conformal field theory in $d$
dimensions using the free flow equation. We first show that an
induced metric defined from the flowed field generally corresponds
to the quantum information metric, called the Bures or Helstrom metric,
if the flowed field is normalized appropriately. We next verify
that the induced metric computed explicitly with the free flow
equation always becomes the AdS metric when the theory is conformal.
We also show that the conformal symmetry in $d$ dimensions converts
to the AdS isometry in $d+1$ dimensions after $d$ dimensional
quantum averaging. This guarantees the emergence of AdS geometry
without explicit calculation.
We next apply this method to non-relativistic systems with anisotropic
scaling symmetries, such as Lifshitz field theories and Schr\”odinger
invariant theories. In consequence we obtain a new hybrid geometry
of Lifshitz and Schr\”odinger spacetimes as a general holographic geometry.
We also show that the bulk hybrid geometry is realized by an
Einstein-Maxwell-Higgs system plus a gauge fixing term for diffeomorphism,
which may be interpreted as a holographic dual of a general
non-relativistic system at the boundary.