Complexity of Quantum-Mechanical Evolutions from Probability Amplitudes

We study the complexity of both time-optimal and time sub-optimal quantum Hamiltonian evolutions connecting arbitrary source and a target states on the Bloch sphere equipped with the Fubini-Study metric. This investigation is performed in a number of steps. First, we describe each unitary Schr\"odinger quantum evolution by means of the path length, the geodesic efficiency, the speed efficiency, and the curvature coefficient of its corresponding dynamical trajectory linking the source state to the target state. Second, starting from a classical probabilistic setting where the so-called information geometric complexity can be employed to describe the complexity of entropic motion on curved statistical manifolds underlying the physics of systems when only partial knowledge about them is available, we transition into a deterministic quantum setting. In this context, after proposing a definition of the complexity of a quantum evolution, we present a notion of quantum complexity length scale. In particular, we discuss the physical significance of both quantities in terms of the accessed (i.e., partial) and accessible (i.e., total) parametric volumes of the regions on the Bloch sphere that specify the quantum mechanical evolution from the source to the target states. Fourth, after calculating the complexity measure and the complexity length scale for each one of the two quantum evolutions, we compare the behavior of our measures with that of the path length, the geodesic efficiency, the speed efficiency, and the curvature coefficient. We find that, in general, efficient quantum evolutions are less complex than inefficient evolutions. However, we also observe that complexity is more than length. Indeed, longer paths that are sufficiently bent can exhibit a behavior that is less complex than that of shorter paths with a smaller curvature coefficient.