Shortest paths for efficient control of indirectly coupled qubits

What is the time-optimal way of realizing quantum operations? Here, we show how important instances of this problem can be related to the study of shortest paths on the surface of a sphere under a special metric. Specifically, we provide an efficient synthesis of a controlled-NOT (CNOT) gate between qubits (spins (1)/(2)) coupled indirectly via Ising-type couplings to a third spin. Our implementation of the CNOT gate is significantly shorter than conventional approaches. The pulse sequences for efficient manipulation of our coupled spin system are obtained by explicit computation of geodesics on a sphere under the special metric. These methods are also used for the efficient synthesis of indirect couplings and of the Toffoli gate. We provide experimental realizations of the presented methods on a linear three-spin chain with Ising couplings.