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Complexity and Time

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.