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Intrinsic geometry of quantum adiabatic evolution and quantum phase transitions

We elucidate the geometry of quantum adiabatic evolution. By minimizing the deviation from adiabaticity we find a Riemannian metric tensor underlying adiabatic evolution. Equipped with this tensor, we identify a unified geometric description of quantum adiabatic evolution and quantum phase transitions, which generalizes previous treatments to allow for degeneracy. The same structure is relevant for applications in quantum information processing, including adiabatic and holonomic quantum computing, where geodesics over the manifold of control parameters correspond to paths which minimize errors. We illustrate this geometric structure with examples, for which we explicitly find adiabatic geodesics. By solving the geodesic equations in the vicinity of a quantum critical point, we identify universal characteristics of optimal adiabatic passage through a quantum phase transition. In particular, we show that in the vicinity of a critical point describing a second order quantum phase transition, the geodesic exhibits power-law scaling with an exponent given by twice the inverse of the product of the spatial and scaling dimensions.