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We analytically determine the minimal time and the optimal control lawsrequired for the realization, up to an assigned fidelity and with a fixedenergy available, of entangling quantum gates ($\mathrm{CNOT}$) betweenindirectly coupled qubits of a trilinear Ising chain. The control is coherentand open loop, and it is represented by a local and continuous magnetic fieldacting on the intermediate qubit. The time cost of this local quantum operationis not restricted to be zero. When the matching with the target gate is perfect(fidelity equal to one) we provide exact solutions for the case of equal Isingcoupling. For the more general case when some error is tolerated (fidelitysmaller than one) we give perturbative solutions for unequal couplings.Comparison with previous numerical solutions for the minimal time to generatethe same gates with the same Ising Hamiltonian but with instantaneous localcontrols shows that the latter are not time-optimal.

Time-optimal Unitary Operations in Ising Chains II: Unequal Couplings and Fixed Fidelity