Compared with many other methods which only give time sub-optimal designs, the quantum brachistochrone equation has a great potential to provide accurate time-optimal protocols for essentially any quantum control problem. So far it has been of limited use, however, due to the inadequacy of conventional numerical methods to solve it. Here, using differential geometry, we reformulate the quantum brachistochrone curves as geodesics on the unitary group. This identification allows us to design a numerical method that can efficiently solve the brachistochrone problem by first solving a family of geodesic equations.
Xiaoting Wang, Michele Allegra, Kurt Jacobs, Seth Lloyd, Cosmo Lupo, and Masoud Mohseni, "Time-optimal quantum control via differential geometry," Proc. SPIE 10118, Advances in Photonics of Quantum Computing, Memory, and Communication X, 101180F (Presented at SPIE OPTO: February 01, 2017; Published: 20 February 2017); https://doi.org/10.1117/12.2256267.
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