We solve the problem of steering a three-levelquantum system from one eigen-state to another in minimumtime and study its possible extension to the time-optimal controlproblem for a general n-level quantum system. For the threelevelsystem we find all optimal controls by finding two types ofsymmetry in the problem: Z2 × S3 discrete symmetry and S1continuous symmetry, and exploiting them to solve the problemthrough discrete reduction and symplectic reduction. We thenstudy the geometry, in the same framework, which occurs inthe time-optimal control of a general n-level quantum system.
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