Quantum Circuits for Incompletely Specified Two-Qubit Operators
  While the question ``how many CNOT gates are needed to simulate an arbitrary two-qubit operator'' has been conclusively answered -- three are necessary and sufficient -- previous work on this topic assumes that one wants to simulate a given unitary operator up to global phase. However, in many practical cases additional degrees of freedom are allowed. For example, if the computation is to be followed by a given projective measurement, many dissimilar operators achieve the same output distributions on all input states. Alternatively, if it is known that the input state is |0>, the action of the given operator on all orthogonal states is immaterial. In such cases, we say that the unitary operator is incompletely specified; in this work, we take up the practical challenge of satisfying a given specification with the smallest possible circuit. In particular, we identify cases in which such operators can be implemented using fewer quantum gates than are required for generic completely specified operators. 
Vivek V. Shende and Igor L. Markov