Simplified Approach to Implementing Controlled-Unitary Operations in a   Two-Qubit System
  We introduce a scheme for realizing arbitrary controlled-unitary operations in a two qubit system. If the 2 \times 2 unitary matrix is special unitary (has unit determinant), the controlled-unitary gate operation can be realized in a single pulse operation. The pulse, in our scheme, will constitute varying one of the parameters of the system between an arbitrarily maximum and a "calculated" minimum value. This parameter will constitute the variable parameter of the system while the other parameters, which include the coupling between the two qubits, will be treated as fixed parameters. The values of the parameters are what we solve for using our approach in order to realize an arbitrary controlled-unitary operation. We further show that the computational complexity of the operation is no greater than that required for a Controlled-NOT (CNOT) gate. Since conventional schemes realize a controlled-unitary operation by breaking it into a sequence of single-qubit and CNOT gate operations, our method is an improvement because we not only require lesser time duration, but also fewer control lines, to implement the same operation. To demonstrate improvement over other schemes, we show, as examples, how two controlled-unitary operations, one being the controlled-Hadamard gate, can be realized in a single pulse operation using our scheme. Furthermore, our method can be applied to a wide range of coupling schemes and can be used to realize gate operations between two qubits coupled via Ising, Heisenberg and anisotropic interactions. 
Preethika Kumar, and Steven R. Skinner
quant-ph