The K-P Problem on Tensor Products of Lie Groups and Time Optimal Control of n Quantum Bits With A Bounded Field

We use properties of sub-Riemannian K -P problems and the optimal synthesis on SU(2) to solve the time optimal control of n non-interacting two level quantum systems (qubits) controlled in parallel with a control field bounded in L2 norm. Every qubit is controlled independently but the total energy of the controls used is bounded. For this system, we describe the reachable set at every time. Moreover, we compute the diameter, that is, the worst case minimum time. The mathematical problem is motivated by the control of two level systems with different resonance frequencies where the control fields are obtained from a single signal having bounded energy.