We solve the problem of steering a three-level quantum system from one eigen-state to another in minimum time and study its possible extension to the time-optimalcontrol problem for a general n-level quantum system. For the three-level system we find alloptimal controls by finding two types of symmetry in the problems: Z2 ×S3 discrete sym-metry and S1 continuous symmetry, and exploiting them to solve the problem throughdiscrete reduction and symplectic reduction. We then study the geometry, in the sameframework, which occurs in the time-optimal control of a general n-level quantum system.
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