  For any quantum algorithm given by a path in the space of unitary operators
we define the computational complexity as the typical computational time
associated with the path. This time is defined using a quantum time estimator
associated with the path. This quantum time estimator is fully characterized by
the Lyapunov generator of the path and the corresponding quantum Fisher
information. The computational metric associated with this definition of
computational complexity leads to a natural characterization of cost factors on
the Lie algebra generators. Operator complexity growth in time is analyzed from
this perspective leading to a simple characterization of Lyapunov exponent in
case of chaotic Hamiltonians. The connection between complexity and entropy is
expressed using the relation between quantum Fisher information about quantum
time estimation and von Neumann entropy. This relation suggest a natural bound
on computational complexity that generalizes the standard time energy quantum
uncertainty. The connection between Lyapunov and modular Hamiltonian is briefly
discussed. In the case of theories with holographic duals and for those reduced
density matrix defined by tracing over a bounded region of the bulk, quantum
estimation theory is crucial to estimate quantum mechanically the geometry of
the tracing region. It is suggested that the corresponding quantum Fisher
information associated with this estimation problem is at the root of the
holographic bulk geometry.
