Quantum Brachistochrone
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We investigate the application of optimal control of a single-qubit coupled to
an ohmic heat bath. For the weak bath coupling regime, we derive a Bloch-Redfield
master equation describing the evolution of the qubit state parameterized by
vectors in the Bloch sphere. By use of the optimal control methodology we
determine the field that generates a single qubit rotation. We use the techniques
of automatic differentiation to compute the gradient for the cost functional. We
consider also the concept of ”Quantum Brachistochrone”. Here the problem
naturally arises of determining the minimal transition time between an initial
state and a final state. The optimal control is of ”bang-bang” type and switches
from the upper to the lower value of the control bounds.